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Systematics of quark-antiquark states and scalar

exotic mesons

V.V. Anisovich

Abstract

The analysis of the experimental data of Crystal Barrel Collaboration on thepp annihilation in flight with the production of mesons in the final state resultedin a discovery of a large number of mesons over the region 1900–2400 MeV, thusallowing us to systematize quark-antiquark states in the (n,M2) and (J,M2)planes, where n and J are radial quantum number and spin of the meson withthe mass M . The data point to meson trajectories in these planes being approx-imately linear, with a universal slope. The sector of scalar mesons is discussedin more detail, where, on the basis of the recent K-matrix analysis, the nonetclassification of quark–antiquark states was performed: in the region below 2000MeV, two scalar nonets are fixed, that is, the basic one and the nonet of thefirst radial excitation. In the scalar sector, there are two states with the isospinI = 0, which are extra ones with respect to the quark–antiquark classification,i.e. exotic states: the broad resonance f0(1200 − 1600) and the light σ-meson.The ratios of coupling constants for hadronic decays to the states ππ,KK, ηη, ηη′

point to the gluonium nature of the broad state f0(1200 − 1600).

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Contents.

1. Introduction.2. Experimental data of the Crystal Barrel Collaboration and systematics of mesonsstates.2.1 Systematics of meson states.2.2 Exotics: the states off quark–antiquark trajectories.3. Scalar-meson sector.3.1 The K-matrix analysis of the (IJPC = 00++)-wave.3.1.1 K-matrix amplitude.3.1.2 Partial amplitude for the 00++ wave: unitarity, analyticity and the problem ofthe left-hand cut.3.1.3 Three-meson production in the reactions of the pp and np annihilation.3.1.4 Peripheral two-meson production in meson–nucleon collisions at high energies.3.2 Classification of scalar bare states.3.3 Overlapping f0-resonances in the region 1200–1700 MeV: the accumulation ofwidths of quark-antiquark states by the glueball.3.4 Evolution of couplings of the 00++-states with channels ππ, ππππ, KK, ηη ηη′

with the onset of the decay processes.3.5 Evaluation of the glueball component in the resonances f0(980), f0(1300), f0(1500),f0(1750) and broad state f0(1200− 1600) based on the analysis of hadronic channels.3.6 The light σ-meson: Is there a pole of the 00++-wave amplitude?3.7 Systematics of scalar states on the (n,M2) and (J,M2) planes and the problem ofbasic multiplet 13P0qq.3.7.1 The K-matrix classification of scalars and qq trajectories in the (n,M2) planes.3.7.2 Systematics of kaons in the (J,M2) plane.3.8 Exotic scalar states, f0(1200− 1600) and f0(300− 500).4. Conclusion.

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1 Introduction

To understand the structure of strong interactions at low and moderate energies is oneof the highest-priority problems of the modern particle physics. In the last decades,great efforts have been paid to develop the strong-interaction theory, the strong QCD,with a considerable progress though without determinate breakthrough. One maybelieve that, in part, this was due to a poor knowledge, up to recent time, of theexperimental situation in meson sector.

To enlighten experimental situation, a number of experiments had been undertakenduring 1990–2000 directed purposefully to the search for new meson resonances, and adetailed investigation of mesons discovered before was carried out.

The Crystal Barrel Collaboration accumulated one of the richest statistics on thepp and np annihilation reactions; the group of physicists from PNPI took part in theanalysis and interpretation of the data by Crystal Barrel Collaboration (a brief reviewof this activity is given in [1]); this survey presents the results of recent investigations.

In 1993–1994, two groups, QMWC (London) and PNPI (St.Petersburg), were analysingmeson spectra obtained in reactions of the pp annihilation at rest. As a result, thescalar-isoscalar resonances f0(1370) and f0(1500) have been discovered [2, 3, 4, 5]; nowthese resonances are actively discussed in connection with the glueball problem. Toestablish the systematics of quark-antiquark states, the study of pp annihilation inflight carried out in 1999-2001 [6] was of a particular importance, for it allowed us toinvestigate the mass region 1900–2400 MeV: a large number of resonances discovered inthis region made it possible to fix the qq trajectories on the (n,M2) and (J,M2) planes[1, 7]. We speak about the systematics of the qq states in Section 2 of this paper.

Scalar mesons play crucial role for understanding the strong QCD. Starting from1995, the PNPI group worked upon the K-matrix fit to the wave (IJPC = 00++) basedon the simultaneous analysis of all data available by that time. The necessity of acombined analysis was dictated by the existence of large interference effects ”resonance–background” as well as the effects associated with the resonance overlapping. In asituation of such a type, only a combined fitting to a large number of reactions allowsone to hope for reliable results. In Section 3, a current understanding of scalar-mesonsector is presented based on recent results of the K-matrix fit to meson spectra [8].

Previous analyses carried out in 1997–1998 [9, 10, 11] were based on the experi-mental data as follows:(1) GAMS data on the S-wave two-meson production in the reactions πp → π0π0n,ηηn and ηη′n at small nucleon momenta transferred, |t| < 0.2 (GeV/c)2 [12, 13];(2) GAMS data on the ππ S-wave production in the reaction πp → π0π0n at largemomentum transfers squared, 0.30 < |t| < 1.0 (GeV/c)2 [12];(3) BNL data on the reaction π−p→ KKn [14];(4) CERN-Munich data on π+π− → π+π− [15];(5) Crystal Barrel data on pp(at rest, from liquid hydrogen)→ π0π0π0, π0π0η, π0ηη[3, 5, 16].

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Now the experimental basis has much broadened, and additional samples of dataare included into the analysis [8] of the 00++ wave as follows:(6) Crystal Barrel data on proton-antiproton annihilation in gas: pp(at rest, fromgaseous hydrogen)→ π0π0π0, π0π0η [17],(7) Crystal Barrel data on proton-antiproton annihilation in liquid: pp(at rest, fromliquid hydrogen)→ π+π−π0, K+K−π0, KSKSπ

0, K+KSπ− [17];

(8) Crystal Barrel data on neutron-antiproton annihilation in liquid deuterium: np(atrest, from liquid deuterium)→ π0π0π−, π−π−π+, KSK

−π0, KSKSπ− [17];

(9) E852 Collaboration data on the ππ S-wave production in the reaction π−p→ π0π0nat the nucleon momentum transfers squared 0 < |t| < 1.5 (GeV/c)2 [18].

In addition to Ref. [9], the reactions of the pp annihilation in gas have been alsoincluded into the analysis [8]. One should keep in mind that in liquid hydrogen thepp annihilation is going dominantly from the S-wave state, while in gas there is aconsiderable admixture of the P -wave, thus giving an opportunity to analyse the three-meson Dalitz plots in more detail. New Crystal Barrel data allowed us to study thetwo-kaon channel with a more confidence as compared to what had been done before.This is undoubtedly important for the conclusion about the quark-gluon content of thescalar–isoscalar f0-mesons under investigation.

Experimental data of the E852 Collaboration on the reaction π−p → π0π0n atplab = 18 GeV/c [18] together with the GAMS data on the reaction π−p → π0π0n atplab = 38 GeV/c [12] give us a solid ground for the study of the resonances f0(980) andf0(1300), for at large momenta transferred to the nucleon, |t| ∼ 0.5 ∼ 1.5 (GeV/c)2,the production of resonances is accompanied by a small background, thus allowing usto fix reliably their masses and widths. This is especially important for f0(1300): inthe compilation [19] this resonance is quoted as f0(1370), with the mass in the interval1200 − 1500 MeV, though experimental data favour rather definitely the mass near1300 MeV.

The K-matrix amplitude determines both the amplitude poles (masses and widthsof resonances) and K-matrix poles (masses of bare states). The K-matrix poles differfrom the amplitude poles in two items:(i) The states corresponding to the K-matrix poles do not contain any componentassociated with the decay processes, i.e. the transitions into real mesons. The absenceof a cloud of real mesons allows us to refer conventionally to these states as bare ones[9, 11, 20].(ii) Due to the transition bare state(1) → real mesons → bare state(2) the observedresonances are mixtures of bare states — in the first place this is related to the f0-mesons. So, for quark systematics, the bare states are primary objects rather than theresonances.

This can be explained with the example of the behaviour of a level in the potentialpicture. Consider a potential well and the levels which correspond to stable states.Then gradually we switch the decay channels on, that is, replace impenetrable wall bypotential barrier — at the beginning this leads to a broadening of levels, but masses

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of states remain almost the same. But when the widths and resonance positions aresuch that resonances overlap, a cardinal reconstruction of the structure of levels oc-curs. Namely, one of resonances accumulates the widths of its resonance-neighboursthus making them comparatively narrow. In this way an intensive mixing of resonancestates takes place, while the positions of masses are shifted in a value of the orderof the resonance width, in 100–200 MeV. The K-matrix amplitude allows us to tracethe transformation of stable levels (bare states) into resonances. The K-matrix am-plitude works with parameters related to stable levels. Multichannel unitarization ofthe amplitude, which is inherent in the K-matrix representation, with the account foranalyticity, allows one to trace the transformation of stable levels into real resonances.A characteristic feature of the K-matrix fit is its ability to reconstruct the picture ofstable levels as well as re-create a real picture of complex masses and partial widths.

The latest K-matrix fit [8] gave us rather definite information on the resonancesf0(980), f0(1300), f0(1500), f0(1750) and broad state f0(1200 − 1600). Relying onthe extracted partial widths for transitions to the channels ππ,KK, ηη, ηη′, one cananalyse the quark-gluonium content of these states. In this way the following propertiesof resonances are to be formulated:

1. f0(980): This resonance is dominantly the qq state, qq = nn sinϕ + ss cosϕ,with a large ss component. Assuming the glueball admixture to be not greater than20%,Wgluonium

<∼ 20%, the hadronic decays give us the following constraints for mixingangle: −95◦ ≤ ϕ <∼ −40◦. Rather large uncertainties in the determination of mixingangle are due to a high sensitivity of coupling constants to a plausible small admixtureof the gluonium. At Wgluonium = 0, hadronic decays provide us with ϕ = −67◦ ± 10◦.

2. f0(1300) (in the compilation [19] this resonance is denoted as f0(1370)): Thisresonance is the descendant of the bare qq state which is close to the flavour sin-glet. The resonance f0(1300) is formed due to a strong mixing with the primarygluonium and neighbouring qq states. The quark-antiquark content of f0(1300), qq =nn cosϕ + ss sinϕ determined from the transitions f0(1300) → ππ,KK, ηη stronglydepends on the admixture of the gluonium component. At Wgluonium

<∼ 30% themixing angle changes, depending on Wgluonium and interference sign, in the inter-val −45◦ <∼ ϕ[f0(1300)] <∼ 25◦; at Wgluonium = 0 the hadronic decays provide usϕ[f0(1300)] = −6◦ ± 10◦.

3. f0(1500): This resonance is the descendant of a bare state with large nn compo-nent. Like f0(1300), the resonance f0(1500) is formed by mixing with the gluonium andneighbouring qq states. The quark-antiquark content, qq = nn cosϕ+ss sinϕ, dependson the admixture of the gluonium: at Wgluonium

<∼ 30% the mixing angle changes, de-pending on Wgluonium, in the interval −20◦ <∼ ϕ[f0(1300)] <∼ 25◦; at Wgluonium = 0 onehas ϕ[f0(1500)] = 11◦ ± 8◦.

4. f0(1750): This resonance is the descendant of the bare state belonging to theradial-excitation nonet 23P1qq, the wave function of which has large ss component.The K-matrix analysis permits the two solutions, with different values of the ss com-ponent. In the first solution, the ss component dominates; in the absence of gluonium

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component, ϕ[f0(1750)] = −72◦ ± 5◦, and if the gluonium admixture does not ex-ceed 30%, then −110◦ <∼ ϕ[f0(1750)] <∼ −35◦. In the second solution, in the absenceof gluonium component, ϕ[f0(1750)] = −18◦ ± 5◦, and with the gluonium admixtureWgluonium

<∼ 30%, −50◦ <∼ ϕ[f0(1750)] <∼ 10◦.

5. f0(1200−1600): The broad state is the descendant of the primary glueball. Theanalysis of hadronic decays of this resonance confirmed its glueball nature: the glueballdescendant has the quark-antiquark component (qq)glueball = (uu+dd+

√λss)/

√2 + λ

[21], which is defined by the probability to create new qq-pairs by the gluon fielduu : dd : ss = 1 : 1 : λ, the suppression parameter for the production of strange quarksbeing in the interval λ ≃ 0.5− 0.8 [22, 23]. In terms of mixing angle for the nn and sscomponents, this means that the glueball descendant must have ϕglueball ≃ 27◦ − 33◦,and just these values for ϕ[f0(1200 − 1600)] have been obtained in all variants of theK-matrix fit [8]. Such a value of mixing angle, ϕ[f0(1200− 1600)] = 30◦± 3◦, does notallow one to determine the admixture of the qq component in the broad state. Thisis due to the fact that the (qq)glueball and gluonium components are coupled to thechannels ππ,KK, ηη and ηη′ in equal proportions.

Referring to the broad state f0(1200− 1600) as a resonance needs some comments.The observed spectra in the channels ππ,KK, ηη, ηη′ demand to introduce a broadbump, and it occurred that this bump behaves in a universal way, thus enabling usto describe it as a resonance state. Indeed, a characteristic feature of the resonance isthe factorization property: the resonance amplitude may be represented in the formgin(s −M2)−1gout, and universal constants gin and gout depend on the sort of initialand final states only. The description of a large number of reactions in [8] agreeswell with the factorization property of the broad-state amplitude. A large width ofthe f0(1200 − 1600) does not allow us to determine reliably its mass, but due to astrong production of the f0(1200− 1600) in a large number of reactions we can, with aconfidence, to find out the ratio of couplings to the channels ππ,KK, ηη, ηη′, that is,to define its content in terms of the qq and gluonium states.

The K-matrix [8] analysis does not point determinedly to the existence of the lightσ-meson which is actively discussed at present (e.g. see [24] and references therein),in particular in connection with the recently reported signals in the ππ spectra of thedecays D+ → π+π+π− (pole in the amplitude at M = (480 ± 40) − i(160 ± 30)MeV [25]), J/Ψ → ππω (pole at M = (390 +60

−36) − i(141 +38−25) MeV [26]), τ → πππν

(pole at M ≃ 555 − i270 MeV [27]). Possible explanation of this discrepancy mayconsist in a strong suppression of the light σ-meson production in the pp annihilationprocesses, like pp → πππ, though, as one may think, there is no visible reason forsuch a suppression. Recall that the statistics in the Crystal Barrel reactions is bytwo orders of magnitude larger than in [25, 26, 27]. Alternative explanation can beassociated with a restricted applicability of the K-matrix approach at small invariantenergy squared s: the K-matrix amplitude does not describe properly the left-handsingularities associated with meson exchanges in the crossing channels. So one maysuppose that theK-matrix analysis does not reconstruct analytical amplitude correctlyat Re s <∼ 4m2

π, i.e. at (ReM)2 − (ImM)2 <∼ 4m2π, (ReM)2 − (ImM)2 <∼ 4m2

π. In

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numerous analyses, including those carried out in the dispersion relation techniquewhere left-hand cuts can be accounted for in one way or another, the pole ascribedto the light σ-meson occurred just at Re s <∼ 4m2

π, e.g. see [28, 29, 30, 31]. In thisconnection let us emphasize that the dispersion-relation N/D analysis [31], where thelow-energy 00++ amplitude was sewed with the K-matrix amplitude [9], leads to thepole at M ≃ 430− i325 MeV.

Therefore, the K-matrix amplitude analysis tells us that in the scalar–isoscalarsector there are two states, the broad resonance f0(1200− 1600) and the light sigma-meson f0(300−500), which are extra ones for the qq-systematics of mesons. In the finalSection 4, the status of these possible exotic states is discussed as well as two morecandidates for exotics — π2(1880) and the (JPC = 1−+) state (recall that qq-systemcannot have such quantum numbers).

2 Experimental data of the Crystal Barrel Collab-

oration and systematics of mesons states

In 1989-1997, the Crystal Barrel Collaboration studied the reactions of the pp-annihilationat rest and in flight at LEAR (CERN), that resulted in the accumulation of a hugestatistics on multi-meson states. However, initially the analysis of experimental datahad not been carried out in due course. The first attempt to analyse pion spectra inthe reaction pp (at rest) → π0π0π0 based on a simplified isobar model led to a wrongidentification of the peak near 1500 MeV; the peak had been identified as the tensormeson AX(1515) [32], and at the same time no scalar states were seen in the ππ spectraover the range 1200–1700 MeV.

In reactions such as pp (at rest) → π0π0π0 the three-particle interaction charac-teristics reveal themselves in full scale, that should be accounted for in the analysisof meson spectra. The method based on the extraction of leading singularities (pole,square root, logarithmic ones, and so on) in the production amplitude of a few parti-cles was developed in the papers [33, 34, 35]. This method had been applied with thepurpose to re-analyse the Crystal Barrel data for the reaction pp (at rest) → π0π0π0,and in the work performed together with the Collaboration members, the resonancesf0(1370) and f0(1500) had been discovered in the region 1300–1500 MeV [2, 3, 4]. Thedata on the reactions pp→ π0π0η and pp→ π0ηη involved into combined analysis en-abled us to discover a group of tensor and scalar-isovector resonances [5]. The massesand widths of these resonances obtained in the latest analysis are shown in Table 1,and the analysed reactions are quoted in Table 2.

A combined fit to data collected in various reactions and experiments is a specificfeature of the method used in the papers [8, 9, 10]. The matter is that, because of thepresence of a considerable reaction background, the resonance does not always revealitself as a peak in meson spectrum: due to interference effects the resonance may appearas a dip in the spectrum, or it may be seen as a ”shoulder”. Including into analysis

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the GAMS data on the two-meson spectra in the reactions π−p→ π0π0n, π−p→ ηηn,π−p → ηη′n [12, 13], BNL data on π−p → K+K−n [14] and CERN–Munich data onπ−p → π+π−n [15] allowed us not only to determine reliably the ratios of differenttwo-meson yields (that is rather important for the resonance classification), but alsoto conclude that the peak in the ηη-spectrum in the reaction π−p → ηηn, which waspreviously claimed to be the resonance G(1590) [13], appeared in fact as a result of theinterference of the broad state f0(1200− 1600) and f0(1500) resonance [36].

Table 1: Resonances discovered in the analysis of Crystal Barrel data. Masses andwidths are presented in accordance with the latest combined analysis [1,8]

Resonance IGJPC Mass, Width,MeV MeV

f0(1300) (or f0(1370) [21]) 0+0++ 1310± 20 280± 30f0(1500) 0+0++ 1495± 6 126± 5f0(1200− 1600) 0+0++ 1400± 200 1200± 400a0(1450) 1−0++ 1520± 25 240± 20a2(1660) 1−2++ 1670+40

−20 310± 40f2(1565) 0+2++ 1580± 6 160± 20

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Table 2: The list of reactions used in the combined analysis [1,8]

Crystal Barrel Two-particle datareactions Reaction Collaborationpp→ π0π0π0 π−p→ π−π+n CERN- Munichpp→ ηπ0π0 π−p→ π0π0n GAMSpp→ ηηπ0 π−p→ ηηn GAMSpp→ π+π−π0 π−p→ ηη′n GAMSpn→ π+π−π− π−p→ KKn BNLpn→ π−π0π0 π−p→ π0π0n E852pp→ π0KSKS

pp→ π0K+K−

pp→ π+K−KS

pn→ π−KSKS

pn→ π0K−KS

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Table 3: Resonances discovered in the analysis of the pp-annihilation reactions in flight[6]. One star means that the resonance had been observed in one reaction only or itreveals itself poorly. Two stars mean that the resonance was seen in two reactions orin one where its contribution dominates. Three stars mark well-established resonancesby using several reactions.

Resonance IGJPC Mass, MeV Width, MeV Status of the stateπ 1−0−+ 2070± 35 310± 80 *π 1−0−+ 2360± 30 300± 80 *a1 1−1++ 2270± 50 300± 70 *π2 1−2−+ 2005± 20 210± 40 *π2 1−2−+ 2245± 60 320± 60 *a2 1−2++ 1950± 40 180± 40 **a2 1−2++ 2030± 20 205± 30 ***a2 1−2++ 2175+80

−30 310± 60 *a2 1−2++ 2255± 20 230± 15 ***a3 1−3++ 2030± 20 150± 20 **a3 1−3++ 2275± 40 150± 20 *a4 1−4++ 2005± 30 180± 30 ***a4 1−4++ 2255± 40 330± 70 **π4 1−4−+ 2250± 15 215± 25 **f0 0+0++ 2105± 15 200± 25 **f0 0+0++ 2320± 30 175± 45 *f2 0+2++ 1920± 40 260± 40 **f2 0+2++ 2020± 30 275± 35 ***f2 0+2++ 2240± 30 245± 45 ***f2 0+2++ 2300± 35 290± 50 **f4 0+4++ 2020± 25 170± 20 ***f4 0+4++ 2300± 25 280± 50 **η2 0+2−+ 2030± 40 190± 40 **η2 0+2−+ 2250± 40 270± 40 ***ω 0−1−− 2150± 20 235± 30 **ω 0−1−− 2295± 50 380± 60 *ω2 0−2−− 1975± 20 175± 25 *ω2 0−2−− 2195± 30 225± 40 *ω3 0−3−− 1960± 30 165± 30 **h1 0−1+− 2000± 20 205± 20 **h1 0−1+− 2270± 15 175± 30 **ρ1 1+1−− 1980± 30 165± 30 **ρ2 1+2−− 1940± 40 155± 40 *ρ2 1+2−− 2225± 35 335± 75 *ρ3 1+3−− 1980± 15 175± 20 **ρ3 1+3−− 2260± 20 200± 30 *ρ4 1+4−− 2240± 25 210± 40 **b1 1+1+− 1970± 40 215± 60 **b1 1+1+− 2210± 50 275± 45 *b3 1+3+− 2020± 15 110± 20 **b3 1+3+− 2245± 50 350± 80 *

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The data by the Crystal Barrel Collaboration discussed above were obtained fromthe pp-annihilation at rest. In addition, the Crystal Barrel Collaboration has a hugestatistics for the events of the pp-annihilation in flight, the antiproton momentumcovering the range 600–1900 MeV. After the Crystal Barrel Collaboration stopped itsactivity in 1999, these data were provided to the PNPI group for further processingand analysis of spectra. In 1999–2001, together with the colleagues from Queen Maryand Westfield College (London) and Rutherford–Appleton Laboratory, the PNPI groupanalysed these data. More than thirty resonances have been discovered in the massrange 1900–2400 MeV [6]; they are shown in Table 3. The discovery of these resonancesallowed us to establish systematics of meson qq-states on the (n,M2)- and (J,M2)-planes [1, 7] (n is radial quantum number of the qq-state with mass M and J is itsspin).

2.1 Systematics of meson states

In Figs. 1 and 2, the trajectories on the (n,M2)- and (I, JPC)-planes are shown forthe states with negative and positive charge parities:

C = − : b1(11+−), b3(13

+−), h1(01+−), ρ(11−−), ρ3(13

−−),

ω/φ(01−−), ω3(03−−) ; (1)

C = + : π(10−+), π2(12−+), π4(14

−+), η(00−+), η2(02−+),

a0(10++), a1(11

++), a2(12++), a3(13

++), a4(14++),

f0(00++), f2(02

++) . (2)

In terms of qq-states, the mesons of the nonets n2S+1LJ fill in the following (n,M2)-trajectories for M <∼ 2400MeV:

1S0 → π(10−+), η(00−+) ; (3)3S1 → ρ(11−−), ω(01−−)/φ(01−−) ;1P1 → b1(11

+−), h1(01+−) ;

3PJ → aJ(1J++), fJ(0J

++), J = 0, 1, 2 ;1D2 → π2(12

−+), η2(02−+) ;

3DJ → ρJ (1J−−), ωJ(0J

−−)/φJ(0J−−), J = 1, 2, 3 ;

1F3 → b3(13+−), h3(03

+−) ;3FJ → aJ (1J

++), fJ(0J++), J = 2, 3, 4 .

Different orbital momenta can form the trajectories with the same J , namely, J =L ± 1. Therefore, the number of such trajectories doubles; these states are (I, 1−−),(I, 2++), and so on. Isoscalar states have two flavour components nn = (uu+ dd)/

√2

and ss, that also results in a doubling of trajectories such as η(00−+), f0(00++), and

so on.

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The trajectories with negative charge parity, C = −, can be defined practicallyunambigously (in Figs. 1 and 2, black circles stand for the observed states [6, 8, 19],while open circles mark the states predicted by trajectories). The trajectories are linearwith a good accuracy:

M2 ≃ M20 + (n− 1)µ2 , (4)

where M0 is the mass of the basic meson with n = 1, and the slope parameter is aboutµ2 ≃ 1.3GeV2. The trajectory slopes for b1(11

+−) and b3(13+−) are slightly lower: for

them µ2 ≃ 1.2GeV2.

In the sector with C = +, the states πJ belong definitely to linear trajectories withµ2 ≃ 1.2GeV2, with an exception for π(140) that is not a surprise, for the pion israther specific state. The sector of the aJ -states with J = 0, 1, 2, 3, 4 demonstrates aneat set of linear trajectories, with the slopes µ2 ≃ 1.15− 1.20GeV2; the same slope isseen for the f2- and f4-trajectories.

For the f0-mesons the trajectory slope is µ2 ≃ 1.3GeV2. Let us emphasize thattwo states do not lay on linear qq-trajectories, namely, the light σ-meson f0(300−500)[19] and the broad state f0(1200 − 1600), this latter has been fixed by the K-matrixanalysis [8, 9, 10].

The picture of the state location on the (n,M2)-plot is complemented by trajec-tories on the (J,M2)-plots: they are shown in Fig. 3. To draw the daughter (J,M2)trajectories, it was important that leading meson trajectories (π, ρ, , a1, a2 and P

′) werewell-known from the analysis of hadronic diffractive scatterings at plab ∼ 5−50 GeV/c.

Pion trajectories, leading and daughter ones, are linear with a good accuracy, seeFig. 3. The other leading trajectories, ρ, η, a1, a2, f2 or P ′, can be also considered aslinear, with a good accuracy:

αX(M2) ≃ αX(0) + α′

X(0)M2 . (5)

The parameters of linear trajectories determined by masses of the qq-states are asfollows:

απ(0) ≃ −0.015 , α′π(0) ≃ 0.72 GeV−2;

αρ(0) ≃ 0.50 , α′ρ(0) ≃ 0.83 GeV−2;

αη(0) ≃ −0.24 , α′η(0) ≃ 0.80 GeV−2;

αa1(0) ≃ −0.10 , α′a1(0) ≃ 0.72 GeV−2;

αa2(0) ≃ 0.45 , α′a2(0) ≃ 0.91 GeV−2;

αP ′(0) ≃ 0.71 , α′P ′(0) ≃ 0.83 GeV−2. (6)

The slopes of the α′X(0) trajectories are approximately the same. The inverse slope

value, 1/α′X(0) ≃ 1.25 ± 0.15GeV2, is of the order of the slope µ2 for the trajectories

on the (n,M2)-plane:1

α′X(0)

≃ µ2 . (7)

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The daughter trajectories for π (Fig. 3a), a1 (Fig. 3b), ρ (Fig. 3c), a2 (Fig. 3d) and η(Fig. 3e) are determined unambigously. On the P ′-trajectories, leading and daughterones (Fig. 3f), there is no room for the states f0(300− 500) and f0(1200− 1600): thisfact stresses once again that these states are superfluous for the qq-systematics, i.e.they are exotic states.

2.2 Exotics — the states which do not belong to the quark-

antiquark trajectories

As is seen, there are two states in the scalar-isoscalar sector which do not belong eitherto the (n,M2) or (J,M2) trajectories; these are the broad state f0(1200−1600), whichwas found in the K-matrix analysis of the 00++-wave [8, 9, 10], and the light σ-meson.

The problem of the light σ-meson is discussed rather long ago [19, 28, 29, 30, 31],yet, until now, there is no common opinion about the existence of this state.

Recently there appeared indications that a signal from σ-meson has been seen inthe reactions D+ → π+π+π− [25], J/ψ → ππω [26] and τ → πππν [27]. Nevertheless,one cannot state that it was a reliable experimental identification of this state, be-cause in the three-particle spectra, in the region of small ππ-masses, an enhancementof the spectra can occur, due to both reflected signals from other channels and therescattering effects associated with the resonance production in other channels (tri-angle singularity effect [34, 35]). To single out these effects rather large statistics isnecessary, comparable with that for the reactions investigated by Crystal Barrel; how-ever, in the works [25, 26, 27] such a level of statistics had not been reached. Letus stress that the effects of reflected signals and those of triangle singularities wereseen, when the Crysrtal Barrel reactions have been analysed, and for certain reac-tions they occurred to be rather important. Indeed, a correct identification of f0(1500)in the reactions pp → π0π0π0, π0ηη [2, 3, 4] became plausible after accounting forthe interference of the decay f0(1500) → ππ, ηη and reflected resonance signals fromother channels; the triangle-diagram effects were also studied by analysing the reac-tions pp→ π0π0π0, π0ηη, see [2, 4], but it occurred that they do not affect the spectradirectly, and they may be effectively described by introducing complex-valued param-eters for resonance production. Still, in certain cases the triangle diagrams cannotbe taken into account in such a simple way: another reaction investigated by Crys-tal Barrel, that is, pp → ηπ+π−π+π− [37], where singular terms are located near thephysical region, provided us with an example of strongly affected spectra. One canexpect noticeable corrections coming from triangle diagramms also in the reactionsD+ → π+π+π−, J/ψ → ππω and τ → πππν, for the effect of triangle diagrams isenhanced just near the small ππ-masses, for more detail see [34, 35].

It is necessary to underline that in the reactions measured by Crystal Barrel thebumps in the ππ-spectra are observed, which may be considered, owing to incorrecttreatment, as an indication to the light σ-meson. The reaction dp(at rest) → π0π0π−

can serve us as an example (see [8], Fig. 5): in the π0π0-spectrum, there is an obvious

13

enhancement over the phase space in the region Mπ0π0 ∼ 400−500MeV. However, theDalitz-plot analysis proved that this enhancement is due to the reflected signal fromρ(1450) → π−π0. The existence of a reflected signal in the D+ → π+π+π− is alsopossible [38].

To summarize, for the experimental discovery of the light σ-meson the data areneeded, which would exceed the present statistics by one–two orders of value, i.e. itmust be the statistics comparable with that of the Crystal Barrel reactions.

In the paper [39], the observation of the resonance π2(1880) was reported; thisstate does not belong to (n,M2)- and (J,M2)-trajectories. Being an extra state forthe qq-systematics, the π2(1880) can be the hybrid: the qqg-system.

3 Scalar-meson sector

In this Section, the results of the K-matrix analyses performed in [8, 9] for the scalarsector are subsequently presented. The sector of scalar mesons is of a particular interest;a variety of opinions exist concerning the properties of states which belong to thissector, e.g., see [11, 40, 41, 42, 43, 44, 45]. The latest analysis [8] is the most detailedinvestigation of the 00++ wave, where the available data have been used in a full scale.

3.1 The K-matrix analysis of the (IJPC = 00++)-wave

In a set of papers [8, 9, 10, 20], the K-matrix analysis of the waves IJPC = 00++,10++, 02++, 12++ had been carried out in the mass range 280–1900 MeV. For thesestates, the masses and widths of resonances had been found. In the scalar–isoscalarsector the following states are seen (see Fig. 4):

00++ : f0(980), f0(1300), f0(1500) ,

f0(1200− 1600), f0(1750) . (8)

The f0(980) is a well-known resonance, its properties and its nature are intensivelydiscussed during several decades. The f0(1300) resonance is denoted in the compilation[19] as f0(1370), however its mass following the most accurate determination, as isstressed above, is near 1300 MeV — so the notation f0(1300) is used here. The f0(1500)resonance had been discovered in [2, 3, 4], now it is a well-established state. A few yearsago, there existed a strong belief that in the region around 1700 MeV a comparativelynarrow state, fJ(1710), was present, with J = 0 or 2. The K-matrix analysis [8, 9,10, 20] points to the existence of f0(1750), with the width Γ ∼ 140 − 300 MeV: theuncertainty in the definition of the width of f0(1750) is due to a bad knowledge of theππππ channel in this mass range and, correspondingly, with two available solutions,with Γ ∼ 140MeV and Γ ∼ 300MeV.

The broad state f0(1200 − 1600), with a half-width 500 − 900 MeV, is definitelyneeded for the K-matrix analysis. In the paper [9] this state had been denoted as

14

f0(1530+90−250): a large error in the definition of the mass is due to the remoteness of the

pole from the real axis (physical region) as well as to the existence of several solutionsgiven by the K-matrix analysis.

A large number of reactions, which were succefully described with the f0(1200 −1600), proved the valididty of a factorization inherent in the resonance amplitude: nearthe pole the amplitude is gin(s −M2)−1gout, where the universal coupling constants,gin and gout, depend on the type of the intial and final states only. A strong productionof the f0(1200 − 1600) in various processes allows one to fix reliably these couplingconstants.

In the scalar–isovector sector, the K-matrix analysis points to the presence of tworesonances:

10++ : a0(980), a0(1520) , (9)

while in the tensor-meson sector the following states are seen:

12++ : a2(1320), a2(1660) ;

02++ : f2(1270), f2(1525), f2(1580) . (10)

3.1.1 K-matrix amplitude

In the K-matrix analysis, the fitting parameters are the K-matrix elements which arerepresented as the sums of pole terms g(n)a g

(n)b /(µ2

n−s) (s ≡M2 is the invariant energysquared of mesons, s ≡M2) and a smooth s-dependent term fab(s). Namely,

Kab =∑

n

g(n)a g(n)b

µ2n − s

+ fab(s) , (11)

where fitting parameters are g(n)a , µn and fab, and the indices a, b refer to the reactionchannels: f0 → ππ,KK, ηη, ηη′ and ππππ. The K-matrix poles are not the amplitudepoles, i.e. they do not correspond to real states. The amplitude in the K-matrixapproach is written as follows:

A =K

1− iρK, (12)

where ρ is the diagonal matrix of phase spaces for the processes under consideration,ρ = diag (ρ1(s), ρ2(s), ...). Because of that, the amplitude poles correspond to the zerosof the determinant:

det∣∣∣1− iρK

∣∣∣ = 0 , (13)

while the K-matrix poles respond to the states with switched-off decay channels. Thestates associated with the K-matrix poles do not contain a cloud of real mesons whichappear during resonance decay: this circumastance allows one to call the K-matrixpoles the ”bare states” [9, 11, 20, 36].

15

3.1.2 Partial amplitude for the 00++ wave: unitarity, analyticity and the

problem of the left-hand cut

The K-matrix representation of partial amplitude takes account of the well-knownfact that at low and moderate energies the inelastic processes are dominantly two-particle ones. Being applied to the 00++ wave this means that, along with the elasticscattering ππ → ππ (threshold at

√s = 280MeV), we have the transitions ππ → KK

(threshold at√s = 2mK), ππ → ηη and ππ → ηη′ (thresholds at

√s = 2mη and√

s = mη +mη′). Besides, in the 00++-wave at√s >∼ 1300− 1400MeV, a considerable

four-pion production is observed, but this process can be also treated, with a goodaccuracy, as a formation of two ρ’s or two effective σ-mesons: ππ → ρρ → ππππ andππ → σσ → ππππ. The amplitude in the K-matrix representation takes into accountcorrectly both unitarity and threshold singularities for the two-meson processes. Inthis way, the K-matrix representation of the amplitude serves us as a right frame forthe correct reconstruction of the amplitude above the ππ threshold (in our case, overthe range 280–1900 MeV).

In the K-matrix amplitude, the threshold singularities are taken into account bytreating the phase spaces as analytical functions of the total energy squared s: abovethe threshold, the two-particle phase space can be represented in a standard form

ρa(s) =√1− (m1a +m2a)2/s, and below threshold it should be expressed through

analytical continuation: ρa(s) = i√(m1a +m2a)2/s− 1. The phase space of the two-

resonance state, of the type of ρρ or σσ, may be expressed in the form which reproducescorrectly the threshold singularities, namely, threshold singularity at

√s = 4mπ and the

singularity in the lower complex half-plane s (on the third sheet) at√s = 2(mρ−iΓρ/2),

that is related to the production of ρρ, see [8, 9, 11] for more detail.

The singularities, which are not explicitly taken into account within the K-matrixapproach, are the so-called left-hand side singularities of the partial amplitude. Thesesingularities are due to the exchange of particles in the crossing channels (i.e. in the t-and u-channels), they determine the interaction forces. In the ππ-scattering amplitude,the nearest left singularity is located at s = 0; this singularity is associated with thetwo-pion exchanges in the t- and u-channels. A large contribution is provided by theρ-meson exchange, that leads to the logarithmic singularity of the partial amplitude ats = 4m2

π −m2ρ ≃ −0.5GeV2. In this very region, the two-pion exchange in the 00++

wave, that corresponds to the effective σ-meson, contributes significantly too. Thecontribution of tensor-meson exchanges is important in the region s ≃ −1.5GeV2.

The problem of a correct account for left-hand singularities becomes even morecomplicated, for the contributions from various exchanges cancel each other at small sto a great extent [46], so the contribution from the left-hand cut depends strongly onthe details of the t- and u-channel exchange mechanism, in particular on the structureof the ρ, σ, f2 → ππ form factors.

Because of uncertainties in choosing the interaction forces, it would be reasonablenot to inflict any hypothesis about the left-hand cut for the ππ-scattering amplitude,

16

but to allow a freedom for the fitting procedure. The K-matrix technique, which wasused in [8, 9], allows one to do that: this technique may be easily generalized in a waythat enables to take account of the left-hand amplitude singularities. This opportunitywas used in [8, 9], according to what the data require; this is an important item, let usdiscuss it in more detail.

By discussing the left-hand cut of the 00++-amplitude, it is sufficient to restrainour consideration by one, the nearest, ππ channel only. In this case, partial amplitudecan be written as follows:

A(s) =N(s)

1−B(s), (14)

where the pion loop diagram, which ensures the unitarity of the partial amplitude, is

B(s) =

∞∫

4m2π

ds′

π

N(s′)ρππ(s′)

s′ − s− i0. (15)

At s > 4m2π, the imaginary part of the loop diagram is ImB(s) = ρππ(s)N(s) (half-

residue in the pole s′ = s), and the real part is the principal value of the integral (15).The real part, ReB(s), does not contain the ππ-threshold singularity; this singularityis defined by the phase space factor in the imaginary part, ρππ(s). So we may re-writethe amplitude A(s) in the form

A(s) =K(s)

1− iρππ(s)K(s), (16)

where in the right-hand half-plane s > 0 the K-matrix block

K(s) =N(s)

1− Re B(s), (17)

may have the pole singularities only, and in the left-hand one, s ≤ 0, it contains a setof left-hand singularities. By singling out the left-hand side singularities in the explicitform, one can write the K-matrix block as follows:

K(s) =∑

i

g2im2

i − s+ f(s) , (18)

where the pole positions, m2i , are given by the equalities:

[1− Re B(s)]s=m2i= 0 , (19)

and f(s) contains all left-hand singularities:

f(s) =N(s)

1− Re B(s)−∑

i

g2im2

i − s=

−∞∫

sL

ds′

π

disc f(s′)

s′ − s. (20)

17

The magnitude sL defines the location of the closest left-hand singularity; for theconsidered ππ-scattering amplitude, sL = 0.

In the physical region, a suitable approximation for the amplitude is given by sub-stituting the spectral integral (20) with the sum:

−∞∫

sL

ds′

π

disc f(s′)

s′ − s−→

∑

n

fns+ sn

, (21)

where sn > −sL. In the multichannel analyses [8, 9], the approximation had been used,when the left-hand cut in the K-matrix terms Kab was fitted to the one-pole term asfollows:

fab(s) → fabs+ s0

. (22)

The parameter s0 in different variants of the fit [8] appeared in the interval 0.5 <∼s0 <∼ 1.5GeV2, that points to a large contribution of the t- and u-channel f2, ρ andσ exchanges. The use of the two-pole approximation for the multichannel K-matrixdeteriorates the convergence of the fitting procedure, for the available experimentaldata are insufficient to fix unambigously a large number of the left-hand-cut parameters.

More fruitful for the reconstruction of analytical structure of the amplitude nears ∼ 0 is the fitting to the 00++ wave in the region s < 1000 MeV, when one mayrestrict oneself by the one-channel approximation. In Section 3.8, we present moredetailed narration of the study of analytical structure of the amplitude at the regions ∼ 0− 4m2

π, with the restoration of the left-hand singularities obeying equations (20)and (21); this topic is related to the light σ-meson problem.

3.1.3 Three-meson production in the reactions of the pp and np annihilation

The K-matrix representation of the amplitude may be applied to the description of theproduction of resonances in the three or more particle production. The uniformity ofthe amplitude representation is important in the combined analysis of the two-particle,like ππ → ππ,KK, and multiparticle, like pp → πππ, πηη, πKK, reactions. The K-matrix approach to multiparticle processes is based on the fact that the denominatorof the K-matrix two-particle amplitude (12), [1− ρK]−1, describes pair interactions ofmesons in the final state as well.

Let us clarify this statement using as an example the amplitude of the pp annihila-tion from the 1S0 level: pp(

1S0) → threemesons. Let the produced mesons be labelledby the indices 1,2,3; then the production amplitude for the resonance with the spinJ = 0 in the channel (1,2) is as follows:

A3(s12) = K(prompt)3 (s12)[1− iρ12K12(s12)]

−1, (23)

where the matrix factor [1 − iρ12K12(s12)]−1 depends on the invariant energy squared

of the mesons 1 and 2 and it coincides with matrix factor of the two-particle amplitude

18

(12). The factor K(prompt)3 (s12) stands for the prompt production of particles 1,2 and

resonances in this channel:

(K

(prompt)3 (s12)

)ab=

∑

n

Λ(n)a g

(n)b

µ2n − s12

+ ϕab(s12) , (24)

where Λ(n)a and ϕab are the parameters of the prompt-production amplitude, and g

(n)b

and µn are defined by the two-meson scattering amplitude, see (12).

It is also appropriate to mention that the description of the two-particle interactionsby using theK-matrix factor (1−ρK)−1 is nothing else than a generalization of the well-known Watson–Migdal formula used for the two-nucleon interactions at low energiesin nuclear reactions with multiple production of nucleons [47].

The whole amplitude for the production of the (J = 0)-resonances is defined by thesum of contributions from all channels:

A3(s12) + A2(s13) + A1(s23) (25)

To account for resonances with the nonzero spin J one needs to substitute in (23):

A3(s12) →∑

J

A(J)3 (s12)X

(J)µ1µ2...µJ

(k⊥12)X(J)µ1µ2...µJ

(k⊥3 ), (26)

where theK-matrix amplitude A(J)3 (s12) is determined by the expression similar to (23),

and X(J)µ1µ2...µJ

stands for meson states with the angular momentum J . The angular-momentum operators depend on the perpendicular components of meson relative mo-menta: k⊥12 and k

⊥3 . Here k

⊥12 is the component of relative momentum of particles 1 and

2, k12 = (k1 − k2)/2, which is orthogonal to the total momentum of particles 1 and 2,p12 = (k1 + k2), namely, (k⊥12p12) = 0; likewise, (k⊥3 p) = 0, where p = k1 + k2 + k3.

For the lowest angular momenta, the operators X(J)µ1µ2...µJ

can be easily written;

for example, for J = 1, 2 we have, up to the normalization factor, X(1)µ (k⊥) ∼ k⊥µ

and X(2)µ1µ2

(k⊥) ∼ (k⊥µ1k⊥µ2

− 13k⊥2g⊥µ1µ2

), where g⊥µ1µ2is the metric tensor in the space

orthogonal to the total momentum. The construction of operators for arbitrary J maybe found in [48].

The covariant operators X(J)µ1µ2...µJ

are determined in the four-momentum space;this is a relativistic generalization of the 3-dimensional angular-momentum operatorsof Zemach [49]. The use of the 4-dimentional operators is suitable for the analysis ofmultiparticle final states, for in this case one does not need numerous Lorentz boosts,which are necessary in the Zemach technique or by using spherical functions.

The amplitude expansion with respect to the states with different angular momentaby using relativistic covariant operators has been carried out for all the reactions pp→threemesons, in the analysis of which the PNPI group took part [2, 3, 4, 5, 6, 8]. Suchan expansion, apart from being simple and compact, as was mentioned above, has anadvantage of taking correct account of kinematical factors, which are significant for the

19

reconstruction of the correct behaviour of the threshold singularities of multiparticleamplitude. In the other orbital-moment decomposition techniques this is a subject ofa special care; in particular, in [50] a special method was suggested to take account ofkinematical factors by using the expansion with respect to spherical functions.

In the paper [48], the operators are also constructed for spins and total angularmomenta in case of fermion and photon systems, which are used for the analysis of thereactions pp→ mesons and γγ → mesons.

The formula (23) serves us for singling out the pole singularities of the amplitudepp(1S0) → threemesons, which are the leading ones. The next-to-leading logarith-mic singularities are related to the rescattering of mesons produced by the decayingresonances (that is, triangle-diagram singularities [34]). The analysis performed in[2, 4] showed that in the reactions pp(at rest)→ π0π0π0, π0π0η, π0ηη, to determineparameters of a resonance produced in the two-meson channels one may not take intoconsideration explicitly the triangle diagram singularities — it is important to accountonly for the complex-valuedness of the prompt production amplitude, that is due tofinal-state interactions (that is, the complexity of parameters Λ(n)

a and ϕab in (24)).Note that it is not a universal rule for the meson production processes in the pp an-nihilation — for example, in the reaction pp → ηπ+π−π+π− [37], the explicit form ofthe triangle singularity is important.

A complete account for unitarity and analyticity in the three-meson productionamplitude is related to the consideration of a full set of meson interactions in the finalstate. In the reactions pp(at rest)→ πππ, ππη, πηη, πKK, the connected system ofthe dispersion-relation N/D equations, with all pair-meson equations accounted for,has been written in [51]. The three-meson production amplitudes being related to eachother by the N/D equations leave less freedom for the fitting than formula (23) and,in principle, they provide more information about meson–meson amplitudes. However,the fitting on the basis of the N/D equations is much more complicated procedure thanthe K-matrix analysis.

3.1.4 Peripheral two-meson production in meson–nucleon collisions at high

energies

The two-meson production reactions πp → ππn, KKn, ηηn, ηη′n at high energiesand small momentum transfers to the nucleon, t, provide us with a direct informa-tion about the amplitudes ππ → ππ, KK, ηη, ηη′, for at |t| < 0.2 (GeV/c)2 thereggeized π exchange dominates the production reactions. At larger |t|, the change ofthe regime occurs: at |t| >∼ 0.2 (GeV/c)2 a significant contribution of other reggeonsbecome plausible (a1-exchange, daughter π- and a1-exchanges). Despite a vague knowl-edge of the exchange structure, the study of the two-meson production processes at|t| ∼ 0.5 − 1.5 (GeV/c)2 looks rather attractive, for at such momentum transfers thebroad resonance vanishes, so the production of the f0(980) and f0(1300) appears practi-cally without background, that is important for finding parameters of these resonances.

20

The amplitude of the peripheral production of two S-wave mesons reads:

(ψN 0RψN)R(sπN , t)KπR(t)

[1− ρK

]−1, (27)

where the factor (ψN 0RψN ) stands for the reggeon–nucleon vertex, and 0R is the spinoperator; R(sπN , t) is the reggeon propagator depending on the total energy squared ofcolliding particles, sπN , and momentum transfer squared t, and the factor KπR(t)[1 −iρK]−1 is related to the block of the two-meson production — one may find the detaileddescription of the amplitude (27) in [8, 52].

The factor KπR(t)[1 − iρK]−1 describes the transitions πR(t) → ππ, KK, ηη, ηη′:

the block KπR(t) is associated with the prompt meson production , and [1 − iρK]−1

is a standard factor for meson rescattrings, see (12). The prompt-production block isparametrized as follows:

(KπR(t)

)πR,b

=∑

n

G(n)πR(t)g

(n)b

µ2n − s

+ fπR,b(t, s) , (28)

where G(n)πR(t) is the f0 production vertex, and fπR,b stands for the background pro-

duction of mesons, while the parameters g(n)b and µn are the same as in the transition

amplitude ππ → ππ, KK, ηη, ηη′, see (12).

At the early stage of the analysis of the S-wave two-pion production, π−p →(ππ)S n, the mechanism of the reggeized-π exchange was suggested both at small andmoderately large |t| [53]. The change of the regime at |t| ∼ 0.2−0.4 (GeV/c)2 has beendescribed by including the effective πP-exchange (P is the pomeron): the amplitude ofthe πP-exchange has another sign than the π-exchange, thus leading to the amplitudezero and, correspondingly, to a zero in the t-distribution at |t| ∼ 0.3 (GeV/c)2, wherethe regime changes. The filling-in of the dips in the t-distributions at |t| ∼ 0.3 (GeV/c)2

can appear due to exchanges of other reggeons, namely, reggeized a1-exchange andcontributions from the daughter trajectories, π(daughter) and a1(daughter). Practically, allthese terms do not interfere in the reaction π−p → (ππ)S n: the π- and a1-exchangesdo not interfere due to different spin structures in reggeon–nucleon vertices, (0π ∼ σ⊥

and 0a1 ∼ σ‖, where σ⊥ and σ‖ are the transverse and longitudinal components of thePauli matrix operating in the spin space of the nucleon). At the same time the contri-butions of leading and daughter trajectories do not interfere practically due to a phaseshift in reggeon propagators (for example, the propagator of the leading π-trajectorymay be considered, with a good accuracy, as real magnitude, while the propagator ofthe π(daughter)-trajectory is nearly imaginary), see [52] for more detail.

The a1 exchange for the reaction πp → (ππ)S n was considered in [10, 54]. In[10], the calculations of the ππ spectra were performed both with the a1 exchange andwithin an effective π exchange. As a result, it occurred that the a1 exchange affectsweakly the parameters of the f0(980) and f0(1300): the matter is that the ππ spectrameasured by GAMS group [12] were averaged over large t-intervals, thence the detailsof the t-distributions are not significant for fixing resonance parameters.

21

New data on the production of the ππ system at |t| ≤ 1.5 (GeV/c)2 [18] gave riseto the discussion about the role of the t exchange in the definition of parameters of thef0(980) and f0(1300). The data of the E852 Collaboration obtained at lower energies(plab = 18GeV/c) as compared to the GAMS energy (plab = 38GeV/c) point definitelyto the fact that the description of the peripheral pion production π−p → (ππ)Sn interms of the leading t exchanges, π and a1, is not complete: the alteration of spectrawith energy in the region |t| ∼ 0.3 − 0.4 (GeV/c)2 and Mππ ∼ 1300MeV proved aconsiderable weight of daughter trajectories: π(daughter) and/or a1(daughter).

Combined analysis of the (ππ)S spectra by GAMS [12] and E852 [18] at |t| ≤1.5 (GeV/c)2 has been performed in [52]. The analysis [52] showed that, though thedata do not allow us to find out the t exchange mechanism unambigously, this cir-cumstance affects weakly the definition of parameters of the produced f0(980) andf0(1300) resonances: in all variants of the fit (including various combinations of theπ(leading), a1(leading), π(daughter), a1(daughter) exchanges, the consideration of the effectiveπP and a1P exchanges at |t| >∼ 0.2 (GeV/c)2, with P being the pomeron, or the Orearmechanism [55]) the parameters of f0(980) and f0(1300) are almost the same. This cir-cumstance allows us to restrain ourselves in the K-matris fit [8] by the two variants ofthe t-channel mechanism, namely, π(leading), a1(leading), π(daughter) or π(leading), a1(leading),a1(daughter).

An opposite statement, according to which the choice of the t-exchange mecha-nism influences signficantly the definition of the parameters of f0(980) and f0(1300),was claimed in [56] (though without performing the fitting to resonance parameters).However, the analysis [52] does not confirm the statements of [56].

3.2 Classification of scalar bare states

The systematics of scalar bare states has been carried out in [57], where bare K-mesonswere found which are needed to fix two quark-antiquark nonets, 13P0qq and 23P0qq.The qq nonet contains two scalar–isoscalar states, f bare

0 (1) and f bare0 (2), scalar–isovector

meson abare0 and scalar kaon Kbare0 . The decay couplings to pseudoscalar mesons for

these four states,

f bare0 (1), f bare

0 (2) → ππ,KK, ηη, ηη′ ,

abare0 → πη, KK ,

Kbare0 → πK, ηK , (29)

are determined, in the leading-order terms of the 1/N -expansion [58], by three param-eters only. They are the common decay coupling g, parameter λ for the probabilityto produce strange quarks in the decay process (in the limit of a precise SU(3)flavoursymmetry we have λ = 1) and mixing angle for ss and nn = (uu + dd)/

√2 compo-

nents in the f0-mesons, ψflavour(f0) = nn cosϕ + ss sinϕi. For the nonet partners,ϕ[f bare

0 (1)]− ϕ[f bare0 (2)] = 90◦.

22

The rigid constraints on the decay couplings of bare states (29) are imposed by thequark-combinatorics relations. The rules of quark combinatorics were first suggestedfor the high-energy hadron production [59] and then extended for hadronic J/Ψ decays[60]. The quark-combinatorics relations were used for the decay couplings of the scalar-isoscalar states in the analysis of the quark-gluonium content of resonances in [61] andlater on in the K-matrix analyses [8, 9, 10, 11, 20, 57].

These constraints being imposed on the decay couplings (29) provide us with anopportunity to fix unambigously the states belonging to the basic nonet [57], also see[8, 9]:

13P0qq : f bare0 (700± 100), f bare

0 (1220± 40) ,

abare0 (960± 30), Kbare0

(1220+50

−150

), (30)

as well as the mixing angle for f bare0 (700) and f bare

0 (1220),

ϕ[f bare0 (700)] = −70◦ ± 10◦ , (31)

ϕ[f bare0 (1220)] = 20◦ ± 10◦ ,

To establish the nonet of the first radial excitation, 23P0qq, is more complicated task.The K-matrix analysis gives us two scalar-isoscalar bare states at 1200–1650 MeV,f bare0 (1230 ± 40) and f bare

0 (1580 ± 40), whose decay couplings (29) obey the relationsappropriate to the glueball. The matter is that the relations between couplings forthe glueball decay, glueball → ππ,KK, ηη, ηη′, and the decay couplings of the quark-antiquark flavour singlet, (qq)singlet → ππ,KK, ηη, ηη′, are almost the same (they areexactly the same in the limit λ = 1). Because of that, it is impossible, by using hadronicdecay couplings only, to distinguish between the glueball and flavour singlet.

Systematics of bare qq-states on the (n,M2)-plane helps us to resolve the dilemmawhich one of these states is the glueball. This systematics (which is discussed in moredetail in Section 3.8) definitely tells us that the state f bare

0 (1580±40) not being on theqq-trajectory is an extra one, so furthermore we accept that this state is the glueball.Indeed,

0++ glueball : f bare0 (1580± 40) . (32)

Lattice calculations [62] agree with this statement: gluodynamical glueball should bein the mass range 1550–1750 MeV.

Having accepted the f bare0 (1580 ± 40) to be non-qq-state, we construct the nonet

23P0qq in a unique way:

23P0qq : f bare0 (1230± 40), f bare

0 (1800± 30) ,

abare0 (1650± 50), Kbare0

(1885+50

−100

). (33)

The kaonic bare states were defined in the K-matrix analysis of the πK-spectrumperformed in [57] on the basis of data [63]. This analysis provided us with a fewsolutions. In (30) and (33) the values of Kbare

0 (1220+50−150) and Kbare

0 (1885+50−100) were

23

used, which were the average ones for these solutions: a large error in the determinationof masses of bare states resulted from the variaty of mass values coming from differentsolutions.

After switching on the decay channels, the bare states (30), (32), (33) turn into realresonances. For scalar–isoscalar states we have, after the decay onset, the transforma-tion as follows:

f bare0 (700± 100) −→ f0(980) ,

f bare0 (1220± 40) −→ f0(1300) ,

f bare0 (1230± 40) −→ f0(1500) ,

f bare0 (1580± 40) −→ f0(1200− 1600) ,

f bare0 (1800± 40) −→ f0(1750) . (34)

This evolution of states is illustrated by Fig. 5, where the shift of amplitude poles intocomlex plane is shown depending on gradual onset of the decay channels. Technically,it is not difficult to switch on/off the decay channels for the K-matrix amplitude: oneshould substitute in the K-matrix elements (11):

g(n)a → ξn(x)g(n)a , fab → ξf(x)fab , (35)

where the parameter-functions for switching on/off the decay channels, ξn(x) and ξf(x),satisfy the following constraints: ξn(0) = ξf(0) = 0 and ξn(1) = ξf(1) = 1, and x variesin the interval 0 ≤ x ≤ 1. Then, at x = 0, the amplitude A turns into the K-matrix,A(x → 0) → K, and the amplitude poles occur on the real axis, that corresponds tothe stable f bare

0 -states. At x = 1, we deal with real resonance; varying x from x = 0 tox = 1 we observe the shift of poles into the complex M-plane.

The x-dependence of ξn(x) and ξf(x) is governed by the dynamics of the statemixing, and the K-matrix solution does not clear up this dynamics. In [64], the mixinghas been modelled within the two-component approach for the decay processes, whenthe resonance widths are due to the transitions f0 → qq and f0 → gg. This modelproved that just the glueball accumulated the widths of neighbouring qq states. Thedominant accumulation of widths by the glueball occurs by virtue of two reasons. First,mutual transitions qq-mesons ↔ glueball are not suppressed within the 1/N expansionrules, e.g. see [11]. Second, the orthogonality of the qq-states suppresses direct mixingof the qq-mesons.

3.3 Ovelapping f0-resonances in the region

1200–1700 MeV: the accumulation of widths of quark-

antiquark states by the glueball

The formation of the broad state is not an accidental phenomenon. The broad state ap-peared as a result of mixture of resonances, due to the transitions f0(1) → real mesons →

24

f0(2). Such transitions in case of overlapping resonances, result in a specific phe-nomenon: when several resonances with common decay channels overlap, one of themaccumulates the widths of neighbouring resonances. So a broad resonance appearstogether with several narrow ones.

Initially, the effect of accumulation of widths had been discussed in nuclear physics[65, 66, 67]. Concerning the scalar 00++-mesons, the accumulation of widths by one ofoverlapping resonances was observed in [20, 36, 68]. In [68, 69], the following schemehas been suggested: the broad state f0(1200 − 1600) is the glueball descendant; thisstate was formed because of the glueball mixing with neighbouring qq-states, thatwas accompanied by the accumulation of widths of neighbouring states by the glueballdescendant. As a result, comparatively narrow states f0(1300), f0(1500), f0(1750) havehad considerable admixtures of the glueball component, while the broad state got alarge qq-component. The quark-antiquark component in the glueball should be closeto the flavour singlet, namely [21]:

(qq)glueball = (uu+ dd+√λss)/

√2 + λ, (36)

where the parameter λ is nearly the same as that in the decay processes, λ ≃ 0.5−0.8.

In meson physics, the accumulation of widths can play a decisive role for the destinyof exotic states which are beyond the qq systematics. Indeed, the exotic state, afterappearing in a set of qq-states, creates a group of overlapping resonances. If thetransition of the exotic state into qq-state is not suppressed (as in case of the glueball,where, according to the 1/N -expansion rules [58], the transition glueball → qq-meson isallowed in the leading-order terms [11, 70]), then just the exotic meson has an advantageto accumulate the widths: the wave functions of neighbouring qq-states are orthogonalto each other but not to the exotic state. Therefore, the existence of the broad statetogether with comparatively narrow ones must serve as a signature of exotics in thismass region [71].

The broad state may play rather constructive role in the formation of the confine-ment barrier. The broad state, after accumulating the widths of resonances–neighboursin the mass scale, plays the role of a locking state. The evaluation of radii of the broadstate f0(1200− 1600) and two narrow neighbouring ones, f0(980) and f0(1300), whichwas performed in [10, 72], tells us that the radius of the broad state is considerablylarger than the radii of f0(980) and f0(1300): this fact agrees well with the assumptionthat f0(1200− 1600) plays the role of a locking state for its resonance-neighbours. Re-cent measurement of the t-distributions in the reaction π−n→ (ππ)S n [18] confirmedrelatively large magnitude of the broad state radius: with the increase of |t| (momen-tum square transferred to the resonance), the broad state vanishes much faster thanf0(980) and f0(1300).

25

Table 4: Coupling constants squared (in GeV2) of scalar–isoscalar resonances decayingto the hadronic channels ππ, KK, ηη, ηη′ and ππππ for different K-matrix solutions.Pole position ππ KK ηη ηη′ ππππ Solutionf0(980)1031− i32 0.056 0.130 0.067 – 0.004 I1020− i35 0.054 0.117 0.139 – 0.004 IIf0(1300)1306− i147 0.036 0.009 0.006 0.004 0.093 I1325− i170 0.053 0.003 0.007 0.013 0.226 IIf0(1500)1489− i51 0.014 0.006 0.003 0.001 0.038 I1490− i60 0.018 0.007 0.003 0.003 0.076 IIf0(1750)1732− i72 0.013 0.062 0.002 0.032 0.002 I1740− i160 0.089 0.002 0.009 0.035 0.168 IIf0(1200− 1600)1480− i1000 0.364 0.265 0.150 0.052 0.524 I1450− i800 0.179 0.204 0.046 0.005 0.686 II

3.4 Evolution of couplings of the 00++-states to channels ππ,ππππ, KK, ηη ηη′ with the onset of decay processes

The K-matrix analysis does not allow one to determine partial widths of the f0-resonances directly. To find out partial widths for the decays f0 → ππ, ππππ, KK,ηη, ηη′ it is necessary to calculate the residues of the amplitude poles corresponding toresonances. Near the resonance, the transition amplitude a→ b (indices a and b standfor the resonance channels ππ, KK, ηη, ηη′, ππππ) takes the form:

Aab ≃ g(n)a g(n)b

µ2n − s

ei(θ(n)a +θ

(n)b

) +Bab . (37)

The pole position defines the mass and width of the resonance µn = Mn − i(Γn/2),

and the real-valued coupling constants to channels, g(n)a and g(n)b , allow us to find

partial widths of resonances. Factorized residues are complex-valued magnitudes, theircomplexity is determined in (37) by the phases θ(n)a and θ

(n)b ; in (37), a smooth, non-

pole, term Bab is also written.

On the basis of the results of the K-matrix fit [8], the decay coupling constantsg(n)a were calculated for f0(980), f0(1300), f0(1500), f0(1750) and the broad statef0(1200 − 1600), and the comparison was done for the obtained values with corre-sponding couplings of bare states, which are predecessors of the resonances underdiscussion.

The values of couplings calculated in [8] are shown in Table 4. The comparisonreveals a significant difference between the decay couplings for bare states and their

26

descendant–resonances. This undoubtedly proves a strong effect of the mixing of qq-states with the glueball: the real resonance is a mixture of these states.

Figure 6 demonstrates the evolution of coupling constants at the onset of the decaychannels: following [73], relative changes of coupling constants are shown for f0(980),f0(1300), f0(1500) and f0(1750) after switching on/off the decay channels (recall thatthe value x = 0 corresponds to the amplitude poles on the real axis and the value x = 1stands for the resonance observed experimentally).

Let us bring our attention to a rapid increase of the coupling constant f0 → KKat the trajectory f bare

0 (700) − f0(980) in the region x ∼ 0.8 − 1.0, where γ2(x =1.0) − γ2(x = 0.8) ≃ 0.2, see Fig. 6a. Actually this increase is the upper limit ofthe possible admixure of the long-range KK component in the f0(980): it cannot begreater than 20%.

3.5 Evaluation of the glueball component in the resonances

f0(980), f0(1300), f0(1500), f0(1750) and broad state f0(1200−1600) based on the analysis of hadronic-decay channels

The evolution of couplings observed in the transition from bare states to real resonancesis due to the mixture of the qq-states with the glueball that is a consequence of thetransitions f0(1) → real mesons → f0(2). One can evaluate the quark-antiquark andglueball components in f0(980), f0(1300), f0(1500), f0(1750) and f0(1200−1600) usingthe rules of quark combinatorics for the decay couplings.

In the leading-order terms of the 1/N -expansion the vertices of hadronic decays ofthe resonances are defined by planar diagrams, the examples of the planar diagramfor the decay of qq-state and the glueball into two mesons are presented in Fig. 7a,b,respectively. In the course of the qq-state decay, the gluons produce a new qq-pair; inthe glueball decay, a subsequent production of two pairs occurs.

For the decay couplings squared for f0 → ππ,KK, ηη, ηη′, the quark-combinatoricsrules, in case when the f0 state is the mixture of the quarkonium and gluonium com-ponents, give us [8, 73]:

g2ππ =3

2

(g√2cosϕ+

G√2 + λ

)2

,

g2KK = 2

g2(sinϕ+

√λ

2cosϕ) +G

√λ

2 + λ

2

,

g2ηη =1

2

(g(cos2Θ√

2cosϕ+

√λ sinϕ sin2Θ) +

G√2 + λ

(cos2Θ+ λ sin2Θ)

)2

,

g2ηη′ = sin2Θcos2Θ

(g(

1√2cosϕ−

√λ sinϕ) +G

1− λ√2 + λ

)2

. (38)

27

The terms proportional to g stand for the transitions qq → two mesons, while thosewith G respond to transitions glueball → two mesons. Accordingly, g2 and G2 areproportional to the probability to find in the considered f0-meson the quark-antiquarkand glueball componets. Recall that the angle ϕ stands for the content of the qq-component in the decaying state, qq = cosϕnn + sinϕ ss, and the angle Θ for thecontents of η and η′ mesons: η = cosΘnn− sinΘ ss and η′ = sinΘnn + cosΘ ss; weuse Θ = 38◦ [74].

One may believe that the decay of the glueball is going in two steps: initially, oneqq pair is produced, then with the production of the next qq pair a fusion of quarksinto mesons occurs. Therefore, at the intermediate stage of the f0 decay, we deal witha mean quantity of the quark-antiquark component, 〈qq〉, which later on turns intohadrons. The equation (38), under the condition G = 0, defines the content of thisintermediate state 〈qq〉 = nn cos〈ϕ〉+ ss sin〈ϕ〉.

As was said above, the K-matrix analysis [8] gave us two Solutions, I and II,which differ mainly by the parameters of the resonance f0(1750). Fitting to the decaycouplings squared for these Solutions leads to the values of 〈ϕ〉 as follows:

Solution I:

f0(980) : 〈ϕ〉 ≃ −68◦ , λ ≃ 0.5− 1.0 , (39)

f0(1300) : 〈ϕ〉 ≃ (−3◦)− 4◦ , λ ≃ 0.5− 0.9 ,

Broad state f0(1200− 1600) : 〈ϕ〉 ≃ 27◦ , λ ≃ 0.54 ,

f0(1500) : 〈ϕ〉 ≃ 12◦ − 19◦ , λ ≃ 0.5− 1.0 ,

f0(1750) : 〈ϕ〉 ≃ −72◦ , λ ≃ 0.5− 0.7 ,

Solution II:

f0(980) : 〈ϕ〉 ≃ −67◦ , λ ≃ 0.6− 1.0 , (40)

f0(1300) : 〈ϕ〉 ≃ (−16◦)− (−13◦) , λ ≃ 0.5− 0.6 ,

Broad state f0(1200− 1600) : 〈ϕ〉 ≃ 33◦ , λ ≃ 0.85 ,

f0(1500) : 〈ϕ〉 ≃ 2◦ − 11◦ , λ ≃ 0.6− 1.0 ,

f0(1750) : 〈ϕ〉 ≃ −18◦ , λ ≃ 0.5 .

In both Solutions, the average values of mixing angle for f0(980) coincide with oneanother, with a good accuracy. Still, one should inderline that equations (38) allow us toget one more magnitude for the mixing angle of the f0(980), namely, 〈ϕ[f0(980)]〉 ≃ 40◦.The fact that the decay constants for f0(980) → ππ and f0(980) → KK accept the

28

solution with 〈ϕ[f0(980)]〉 ≃ 40◦ was underlined in [75]. However, this value of mixingangle does not suit the classification of bare states provided by the K-matrix solutions.Indeed, the f0(980) is the descendant of the bare state f

bare0 (700±100) which is close to

the flavour octet. The evolution of coupling constants (see Fig. 6) tells us that f0(980)by its content remains close to its predecessor, f bare

0 (700 ± 100). Because of that, inEqs. (39) and (40) only solutions with 〈ϕ[f0(980)]〉 ≃ −67.5◦ are kept.

The values of average mixing angles for f0(1300) are stable negative for both Solu-tions I and II, they differ slightly, so we may accept 〈ϕ[f0(1300)]〉 = −10◦ ± 6◦.

Also the mean mixing angle for the f0(1500) does not differ noticeably for SolutionsI and II, so we may adopt 〈ϕ[f0(1500)]〉 = 11◦ ± 8◦.

For the f0(1750), Solutions I and II provide different mean values of mixing an-gle. In Solution I, the resonance f0(1750) is dominantly ss system; correspondingly,〈ϕ[f0(1750)]〉 = −72◦ ± 5◦. In Solution II, the absolute value of mixing angle is muchless, 〈ϕ[f0(1750)]〉 = −18◦ ± 5◦.

For the broad state, both Solutions give proximate values of mixing angle, namely,〈ϕ[f0(1200 − 1600)]〉 = 30◦ ± 3◦. This magnitude favours the opinion that the broadstate can be treated as the glueball descendant, because such a value of the mean

mixing angle corresponds to ϕglueball = sin−1√λ/(2 + λ) at λ ∼ 0.50− 0.85.

Let us emphasize that the coupling magnitudes for the f0-resonances found in [8] donot provide us any alternative variants for the glueball descendant. Indeed, the valuewhich is the closest to the ϕsinglet is the limit value of the mean angle for f0(1500)in Solution I: 〈ϕ[f0(1500)]〉 = 19◦. Such a magnitude being used for the definition ofϕglueball corresponds to λ = 0.24, but this suppression parameter is much lower thanthose observed in other processes: for the decaying processes we have λ = 0.6 ± 0.2[11, 22], while for the high-energy multiparticle production it is λ ≃ 0.5 [23]. In thisway, the quark combinatorics points to the one candidate only, that is, the broad statef0(1200− 1600); we come back to this important statement later on.

Generally, the formulae (38) allow us to find ϕ as a function of the coupling constantratio G/g for the decays glueball → mesons and qq-state → mesons. The results ofthe fit for f0(980), f0(1300), f0(1500), f0(1750) and the broad state f0(1200 − 1600)are shown in Fig. 8.

First, consider the results for f0(980), f0(1300), f0(1500), f0(1750) shown in Fig.8a for Solution I and in Fig. 8c for Solution II. The bunches of curves in the (ϕ,G/g)-plane demonstrate correlations between mixing angle values and the G/g ratios, forwhich the description of couplings given in Table 4 is satisfactory. A vague dissipationof curves, in particular noticeable for f0(1300) and f0(1500), is due to the uncertaintyof λ in Eqs. (39) and (40).

The correlation curves in Fig. 8a,c allow one to see, on a qualitative level, towhat extent the admixture of the gluonium component in f0(980), f0(1300), f0(1500),f0(1750) affects the quark-gluonium content, qq = nn cosϕ+ ss sinϕ determined from

29

hadronic decays. The magnitudes g2 and G2 are proportional to the probability to findout, respectively, the quarkonium and gluonium components, Wqq and Wgluonium in aconsidered resonance:

g2 = g2qqWqq , G2 = G2gluoniumWgluonium . (41)

The results of the K-matrix fit obtained in [8] tell us that the coupling constants g2qqand G2

gluonium are of the same order of magnitude (also see the discussion in [11, 64, 70]),therefore we accept as a qualitative estimate:

G2/g2 ≃Wgluonium/Wqq . (42)

The figures 8a,c show the following permissible scale of values ϕ for the resonancesf0(980), f0(1300), f0(1500), f0(1750), after mixing with the gluonium component.

Solution I:

Wgluonium[f0(980)] <∼ 15% : −93◦ <∼ ϕ[f0(980)] <∼ −42◦, (43)

Wgluonium[f0(1300)] <∼ 30% : −25◦ <∼ ϕ[f0(1300)] <∼ 25◦ ,

Wgluonium[f0(1500)] <∼ 30% : −2◦ <∼ ϕ[f0(1500)] <∼ 25◦ ,

Wgluonium[f0(1750)] <∼ 30% : −112◦ <∼ ϕ[f0(1750)] <∼ −32◦ .

Solution II:

Wgluonium[f0(980)] <∼ 15% : −90◦ <∼ ϕ[f0(980)] <∼ −43◦, (44)

Wgluonium[f0(1300)] <∼ 30% : −42◦ <∼ ϕ[f0(1300)] <∼ 10◦ ,

Wgluonium[f0(1500)] <∼ 30% : −18◦ <∼ ϕ[f0(1500)] <∼ 23◦ ,

Wgluonium[f0(1750)] <∼ 30% : −46◦ <∼ ϕ[f0(1750)] <∼ 7◦ .

The ϕ-dependence of G/g is linear for f0(980), f0(1300), f0(1500), f0(1750). An-other type of the correlation takes place for the state which is the glueball descendant:the correlations curves for this case form in the (ϕ,G/g)-plane a typical cross. Justthis cross appeared for the broad state f0(1200−1600) for both Solutions I and II, seeFig. 8b,d.

The appearance of glueball cross by correlation curves in the (ϕ,G/g)-plane is dueto the formation mechanism of the quark-antiquark component in the gluonium state:in the transition gg → (qq)glueball the state (qq)glueball is fixed by the value of λ. So theglueball descendant is the quarkonium-gluonium composition as follows:

gg cos γ + (qq)glueball sin γ ,

where(qq)glueball = nn cosϕglueball + ss sinϕglueball ,

30

and ϕglueball = tan−1√λ/2 ≃ 26◦ − 33◦ for λ ≃ 0.50− 0.85. The ratios of couplings for

the transitions gg → ππ,KK, ηη, ηη′ are the same as for the quarkonium (qq)glueball →ππ,KK, ηη, ηη′, so the study of hadronic decays only do not permit to fix the mixingangle γ. This property – similarity of hadronic decays for the states gg and (qq)glueball– implies a specific form of the correlation curve in the (ϕ, g/G)-plane: the gluoniumcross. Vertical component of the gluonium cross means that the glueball descendant hasa considerable admixture of the quark-antiquark component (qq)glueball. Horizontal lineof the cross corresponds to the dominant gg component. The value of λ which affectsthe cross-like correlation on the (ϕ, g/G)-plane is denoted from now on as λglueball. ForSolution I, we have λglueball = 0.55, while for Solution II λglueball = 0.85.

At not a large shift of λ from its mean value λglueball, the coupling constantsf0(1200 − 1600) → ππ,KK, ηη, ηη′ can be also described, with a reasonable accu-racy, by Eq. (38); in this case correlation curves on the (ϕ, g/G)-plane take the formof hyperbola. Shifting the value of λ in |λ − λglueball| ∼ 0.2 breaks the description ofcouplings of the broad state by formulae (38).

The cross-type correlation on the (ϕ, g/G)-plane in the description of coupling con-stants f0 → ππ,KK, ηη, ηη′ by formula (38) is a characteristic signature of the glueballor glueball descendant. And vice versa: the absence of the cross-correlation shouldpoint to the quark-antiquark nature of resonance. Therefore, the K-matrix analy-sis proves definitely that f0(1200 − 1600) is the gluonium descendant, while f0(980),f0(1300), f0(1500), f0(1750) cannot pretend to be the glueballs.

The analysis proves that f0(1300), f0(1500) are dominantly the nn-systems. Still,in Solution II the qq component of the resonance f0(1300) may contain rather large sscomponent in the presence of the 30% gluonium admixture in this resonance. As to thef0(1500), the mixing angle 〈ϕ[f0(1500)]〉 in the qq component may reach 25◦ at G/g ≃−0.6 (Solution I) that is rather close to ϕglueball. However, in this case the descriptionof coupling constants g2a (Table 4) is attained as an effect of the strong destructiveinterference of the amplitudes (qq) → two pseudoscalars and gg → two pseudoscalars.This fact tells us that one cannot be tempted to interprete f0(1500) as the glueballdescendant.

3.6 The light σ-meson: Is there a pole of the 00++-wave am-

plitude?

Effective σ-meson is needed in nuclear physics as well as in effective theories of thelow-energy strong interactions — and such an object exists in a sense that there existsrather strong interaction, which is realized by the scattering phase passing throughthe value δ00 = 90◦ at Mππ ≃ 600 − 1000 MeV. In the naive Breit–Wigner-resonanceinterpretation, this would correspond to an amplitude pole; but the low-energy ππamplitude is a result of the interplay of singular contributions of different kinds (left-hand cuts as well as poles located highly, f0(1400−1600) included) , so a straightforwardinterpretation of the σ-meson as a pole may fail.

31

The question is whether the σ-meson exists as a pole of the 00++-wave amplitude.(See also [76], where this problem was particularly underlined.) However, until nowthere is no definite answer to this question, though this point is crucial for mesonsystematics.

The consideration of the partial S-wave ππ amplitude, by accounting for left singu-larities associated with the t- and u-channel interactions, favours the idea of the pole atRe s ∼ 4m2

π. The arguments are based on the analytical continuation of the K-matrixsolution to the region s ∼ 0− 4µ2

π [31].

In [31], the ππ-amplitude of the 00++ partial wave was considered in the region√s < 950MeV. The account for the left-hand singularities had been done within the

method described in Section 3.10.2, namely, the K-matrix amplitude was fitted in theform

K(s) =6∑

n=1

fns+ sn

+ f +g2

M2 − s, (45)

where fn, f, sn, g2,M2 are parameters. The left-hand cut was fitted to six pole terms;

the pole at s =M2 (M ∼ 900 MeV) corresponds to δ00 = 90◦. The fitting was performedto the low-energy scattering phases, δ00, at

√s < 450 MeV, and the scattering length,

a00. In addition at 450 ≤ √s ≤ 950MeV the value δ00 was sewn with those found in

the K-matrix analysis [9]: from this point of view the solution found in [31] may betreated as analytical continuation of theK-matrix amplitude to the region s ∼ 0−4m2

π.The analytical continuation of the K-matrix amplitude of such a type accompanied bysimultaneous reconstruction of the left-hand cut contribution provided us with thecharacteristics of the amplitude as follows. The amplitude has a pole at

√s ≃ 430− i325 MeV , (46)

the scattering length,a00 ≃ 0.22m−1

π , (47)

and the Adler zero at √s ≃ 50 MeV . (48)

The errors in the definition of the pole in solution (46) are large, and unfortunately theyare poorly controlled, for they are governed, in the main, by uncertainties when left-hand singularities are restored. As to experimental data, the position of pole is rathersensitive to the scattering length value, which in the fit [31] was taken in accordance tothe paper [77]: a00 = 0.26± 0.06m−1

π . As one can see, the solution [31] requires a smallscattering length value: a00 ≃ 0.22m−1

π . New and much more precise measurements ofthe Ke4-decay [78] provided a00 = (0.228± 0.015)m−1

π , that agrees completely with thevalue (47) obtained in [31]. Such a coincidence favours undoubtedly the pole position(46).

So, the N/D-analysis of the low-energy ππ-amplitude sewn with the K-matrixone [9], provides us with the arguments for the existence of the light σ-meson. In aset of papers, by modelling the left-hand cut of the ππ-amplitude (namely, by using

32

interaction forces or the t- and u-channel exchanges), the light σ-meson had been alsoobtained [28, 29, 30], but the mass values are widely scattered, e.g. in [79] the pole hasbeen obtained at essentially larger masses,

√s ∼ 600− 900MeV.

The observation of the light σ-meson is aggravated by the existence of the Adler zeroin the ππ-amplitude near the ππ-threshold. Therefore, as was emphasized above, forthe reliable determination of the σ-meson it is necessary to study the ππ-productionin the annihilation and decay reactions (of the type of pp → πππ and D → πππ),where the Adler zero is absent. However, one should emphasize again, in the pp-annihilation, where the statistics is rather high, the σ-meson is not seen, while in thereaction D+ → π+π−π+ [25] the statistics is not sufficient for the reliable analysis ofthe low-energy ππ-spectrum.

By discussing the search for the light σ-meson in the three-particle reactions, it isnecessary to accentuate a requirement, had it been fulfilled, one can confidently confirmthe discovery of this resonance. I mean the rescattering of pions in the final state. Thematter is that the fitting to the near-threshold state, that is actually the σ-meson,should be carried out under the correct account for the unitarity at small masses ofthe ππ-system, while the contribution of crossing channels (for example, resonanceproduction) leads to the violation of unitarity. The account for rescatterings in theππ-channel reconstructs the two-particle unitarity. Besides, the rescatterings restorelogarithmic singularities of the three-particle amplitude, that may affect significantlythe region of small ππ masses, in particular, in the presence of heavy resonances. Theeffects of pion rescatterings were studied in [4], where the combined analysis of theDalitz-plot has been done for the reactions pp(at rest)→ π0π0π0, π0ηη — it appearedthat they act rather strongly upon the region of small ππ-masses, though they are notimportant for the analysis of resonances at 1300–1500 MeV, that was the main goal ofthe investigation performed in [4]. An opposite point of view, namely, the statementabout the possibility to single out the signal from #-meson without taking account ofrescatterings, was claimed in [80].

Concerning the problem of search for the light σ-meson in the three-particle reac-tions, an important point should be emphasized, which being fulfilled would allow usto speak with confidence about reliable determination of this state. This is rescatteringof pions in the final state. The matter is that fitting to a near-threshold state, such asσ-meson, should be carried out with correct account for unitarity at small ππ-masses;at the same time the contribution of crossing channels (for example, the production ofresonances) violates the unitarity. The rescatterings in the ππ-channel being accountedfor restore the two-particle unitarity near the ππ-threshold. Besides, rescatterings leadto logarithmic singularities of the three-particle amplitude, which in the presence ofheavy resonances affect significantly the low-mass region. The effect of pion rescat-terings were considered in [4] at simultaneous analysis of Dalitz-plots in the reactionspp(at rest) → π0π0π0, π0ηη: it occurred that they affect strongly the low-mass regionthough being not important for the analysis of resonances at 1300–1500 MeV — themain scope of investigation in [4]. An opposite viewpoint, namely, one can reliablydetect the σ-meson signal without accounting for ππ-rescatterings, is presented in [80].

33

3.7 Systematics of scalar states on the (n,M2)- and (J,M2)-

planes and the problem of basic multiplet 13P0qq

The figures 1,2 and 3 demonstrate linear behaviour of the qq-meson trajectories in the(n,M2)- and (J,M2)-planes for a variaty of states. In this way, it would be instructiveto juxtapose the nonet classification given by the K-matrix analysis [8, 9, 57] with thesystematics of the qq-states in the (n,M2)- and (J,M2)-planes.

3.7.1 TheK-matrix classification of scalars and qq-trajectories in the (n,M2)-planes

Consider the variant which follows directly from the K-matrix calssification of barestates: let us accept that the light σ-meson does not reveal itself as the amplitude poleor, if it exists, it is an extra state for the qq-systematics. For this case, the location ofscalar states (f0, a0, K0) on the (n,M2) trajectories is shown in Fig. 9.

Figure 9a demonstrates the trajectories for the resonance states 00++, 9++ and 120+,

while in Fig. 9b the trajectories for corresponding bare states are shown (recall thatthe doubling of the f0-trajectories occurs due to the existence of two components, ssand nn). It is seen that the trajectory slopes for bare and real states are nearly thesame (in Fig. 9 the trajectory slope is µ2 = 1.3GeV2).

The state f bare0 (1580 ± 50) certainly does not belong to any of f bare

0 -trajectoriesshown in Fig. 9b, the trajectories of the real f0-states does not need the state f0(1200−1600) too. This is natural, provided the f bare

0 (1580± 50) is the glueball and f0(1200−1600) the glueball descendant.

Gluodynamical lattice calculations tell us that the lightest scalar glueball is locatedin the mass region 1550–1750 [62]. There are also other arguments in favour of justin this mass region one may encounter the gluonium: the estimate of the mass of soft(or effective) gluon points to the gluon mass mgluon ∼ 700 − 1000MeV. Experimentalestimations of the effective-gluon mass are based on the study of hadronic spectra inradiative decays J/ψ → γ + hadrons and Υ → γ + hadrons [81, 82]; they give forthe gluon mass the value 800–1000 MeV. The close values were obtained within modelcalculations of the quark-gluon interactions in the soft region giving the value 700–800 MeV for the effective-gluon mass [83, 84]. The lattice calculations of the effectivegluon agree reasonably with these estimates: accordingly, mgluon ≃ 700MeV [85]. Itis natural to believe that the mass of the lightest scalar glueball is of the order of thedouble effective gluon mass, Mglueball ∼ 2mgluon.

One should pay attention to the fact that the resonances f0(980), f0(1500) andf0(1300), f0(1750) lay neatly on linear trajectories, with a slope which nearly coincideswith slopes of isovector trajectories: ρJ , aJ and πJ . This circumstance argues in favourof the admixture of the glueball components not leading to strong violation of thelinear behaviour of trajectories, at least for the f0-meson sector.

34

It would be pertinent to recall that in the K-matrix analyses [8, 9] three Solutionshave been found denoted as I, II-1 and II-2. Solutions II-1 and II-2 occurred to beclose to each other in resonance characteristics; yet, in Solution II-1 the f bare

0 (1580)is the qq-state, and the glueball mass is around 1250 MeV. We reject this Solution,for it contradicts both the linearity of f bare

0 -trajectories and gluodynamical calculationresults.

Therefore, from the point of view of the classification of states on the (n,M2)-trajectories, the scheme where the f bare

0 (700 ± 100) and its descendant, f0(980), arethe lightest states in the 13P0qq nonet looks self-consistent. This scheme agrees withthe coupling constants for the transitions f bare

0 , abare0 , Kbare0 → ππ,KK, ηη, ηη′ [8, 11,

57]. Moreover, model calculations of masses of the qq-mesons also point to a possiblelocation of the 13P0qq-nonet in the range 900–1300 MeV [86]. Still, by considering thenonet classification, the question arises about large ss-component in the lightest qqstate: 67% in f bare

0 (700± 100) and (45− 75)% in f0(980). But it would be appropriateto recall that we have already faced similar situation in the pseudoscalar sector: theη-meson also contain a large ss-component, about 40%. Both states, η-meson andf bare0 (700 ± 100), are close to the flavour octet, and strong interaction in the flavouroctet states may cause the large ss component in light mesons. In other words, thevalue of the ss component in lightest mesons depends on the structure of the short-range forces, and to determine the structure of forces is the goal of the analysis andthe motivation to perform the systematics of meson states.

So the root of the matter is whether the σ-meson, if it exists, is the standardqq-state, a partner to f0(980). Then the nonet of scalar mesons looks as follows:f0(300−500), f0(980), a0(980) and the low-lying κmeson, κ(800−1000). The argumentagainst this scheme is the systematics of the K-mesons in the (J,M2) plane, that isdiscussed below.

3.7.2 Systematics of kaons in the (J,M2) plane

The existence of the scalar κ-meson, with the mass 800-1000 MeV, is under discussionas long as the discussion of σ-meson, e.g. see [24] and references therein. The low-energy data on the πK-scattering are poor, and this is the main reason of uncertaintyin the κ-meson problem. The K-matrix analysis of the πK-amplitude [57] does notsupply us with unambigous answer, for there exist solutions without amplitude pole at800-1000 MeV but there are also solutions with the pole at ∼ 1000MeV [57]. The K-matrix analysis of the high-statistics Crystal Barrel data on the reactions pp→ KKπand np→ KKπ does not require the κ-meson [8].

Let us turn to the systematics of kaon states on the (J,M2) plane: such systematicsof kaons brings additional argumets to the discussion of whether σ and κ mesons arethe standard qq states, members of the 13P0qq multiplets, or not.

In the kaonic sector, the experimental data are poor, and a complete picture of thekaon disposition on the trajectories with JP = 0−, 2−, 4−, ... and JP = 1+, 3+, 5+, ...

35

cannot be unambigously reconstructed: in Fig. 10a,d a conventional picture is pre-sented for the location of these trajectories. More definite is the situation in thesectors with JP = 0+, 2+, 4+, ... and JP = 1−, 3−, 5−..., see Fig. 10b,c: here we knowthe states laying on the leading and first daughter trajectories — and just on thesetrajectories the κ meson should lay, provided it is the standard qq-state.

In the sectors with JP = 0−, 2−, 4−, . . . and JP = 1+, 3+, 5+, . . . (Fig. 10a,d) thedoubling of trajectories takes place because of the presence of two spin states: forexample, two states with JP = 1+ are defined by the quantum numbers (L = 1, S = 1)and (L = 1, S = 0); likewise, two states JP = 2− are formed by two sets of quantumnumbers, (L = 2, S = 1) and (L = 2, S = 0). Figures 10a,d represent a conjecturaldisposition of trajectories and demonstrate how many unknown states (open circles)are present in the groups JP = 0−, 2−, 4− and JP = 1+, 3+, 5+.

More reliably determined are the trajectories for the groups with JP = 0+, 2+, 4+

(Fig. 10b) and JP = 1−, 3−, 5− (Fig. 10c), where, as was said above, the leading andfirst daughter trajectories are fixed firmly. The leading and first daughter trajectories inthese groups are degenerate: the statesK2+(1430),K4+(2045) andK

∗(890),K3−(1780),K5−(2380) lay on the common leading trajectory αleading(J) ≃ 0.25 + 0.90M2 (theslope is in GeV units), and the states K0+(1430), K2+(1980) and K

∗(1680) lay on thecommon trajectory αdaughter(J) ≃ −2.0 + 0.90M2, where JP = 0+, 1−, 2+, 3−.

Figure 10b demonstrates also the location of the κ meson: one can see that thisstate does not lay either on leading or daughter trajectories. In order to put the κmeson onto the qq trajectory, the existence of the 2+ kaon with the mass ∼ 1600 MeVis needed, but the analysis [63] does not point to the presence of tensor resonancein this mass region. The fact that there is no room for the κ meson on the (J,M2)qq-trajectories is the argument against the construction of the lowest scalar nonet byusing the states f0(300− 500), f0(980), a0(980) and κ(800− 1000), that is actually anindication that the κ meson does not exist.

3.8 Exotic scalar states, f0(1200− 1600) and f0(300− 500)

Two scalar states, the broad resonance f0(1200−1600) and the light σ-meson f0(300−500), remain beyond the considered here qq classification that is based on the K-matrixanalysis [8, 9]; they should be treated as exotic states.

The K-matrix analysis gives us an unambigous interpretation of the f0(1200 −1600). This state is the descendant of a pure glueball which, by accumulating ofwidths of neighbouring scalar–isoscalar states, turned into the broad resonance. Allcomparatively narrow resonances from the region ∼ 1500MeV (they are f0(1300),f0(1500) and f0(1750)) fit well the qq-trajectories in the (n,M2) plane, so just thebroad resonance f0(1200 − 1600) is an extra one from the point of view of the qq-systematics.

The fact that the glueball turned into the broad state allows us to say that the

36

glueball is ”melted”. The idea that the bound state of gluons can disappear due tostrong interaction in the soft region was brought out rather long ago [87]. However, the”melting” observed in the K-matrix analysis is a specific one: the transformation ofthe glueball into the broad state occurs as a result of the decay processes (transitionsof resonances into real mesons) that takes place at large distances, of the order orgreater than Rconfinement. Besides, the amplitude pole associated with f0(1200− 1600)did not go far away from the physical region: the resonance half-width, that definesthe imaginary value of mass related to the pole, is of the order of 500–1000 MeV, sothe pole occurs inside the hemicircle, where the analysis of experimental data allowedus to reconstruct the analytical amplitude (supposing the threshold singularities becorrectly taken into account).

The transition of the lightest scalar glueball into the broad resonance f0(1200−1600)gives rise to a number of questions. The glueball tranformation into the broad reso-nance, is it a unique event (in a sense, an occasional event) or is this a commonphenomenon for exotic states? The resonance f0(1200 − 1600), after having accumu-lated the widths of neighbouring qq-states plays, with respect to them, the role of alocking state, does this lead to the increase of the proper size of f0(1200−1600)? Thesequestions were put in [10, 71]; still, to have a veracious response one needs much moreaccurate data. Qualitative evaluation of the radius of f0(1200−1600) had been carriedout in [10, 71, 72], on the basis of the GAMS data for theMππ-distributions at differentmomenta transferred to the nucleon in the reaction π−p → nπ0π0 at plab = 38GeV/c[12]. According to this estimate, the broad state is much more loosely-bound systemthan its neighbours-resonances. Recent measurements performed by the E852 Collabo-ration [18] support the fact that comparatively narrow resonances f0(980) and f01300)are more compact than the broad state f0(1200 − 1600); the discussion of these datacan be found in [8, 52].

The situation with the light σ-meson, f0(300 − 500), is less definite as concern itsexperimental status and the understanding of its nature. The nature of the σ-meson,if it exists as an amplitude pole, is rather enigmatic. The light σ-meson is hardlythe glueball-like formation. Also it is difficult to imagine the light σ-meson to be thestandard qq-system: the arguments against such an interpretation were formulated inSection 3.8.

It was suggested in [88] that the existence of the light σ-meson may be due to a sin-gular behaviour of the forces between quark and antiquark at large distances (in quarkmodels they are conventionally called ”confinement forces”). The scalar confinementpotential, which defines the spectrum of the qq-states in the region 1000–2000 MeV,behaves at large hadronic distances as V

(c)confinement(r) ∼ αr, where α ≃ 0.19GeV2. In

the momentum representation, such a growth of the potential is associated with thesingular behaviour at small q:

V(c)confinement(q) ∼ 1

q4. (49)

In the colour space, the main contribution comes from the component c = 8, i.e.

37

the confinement forces should be the octet ones. The question that is crucial forthe structure of σ-meson is as follows: is there a component with the colour singlet,V

(1)confinement(q), in the singular potential (49)? If the singular component with c = 1

exists, then it must reveal itself in hadronic channels as well, that is, in the ππ-channel.In hadronic channels, this singularity should not be exactly the same as in the colouroctet ones, because the hadronic unitarization of the amplitude (which is absent inthe channel with c = 8) should modify somehow the low-energy amplitude. One maybelieve that, as a result of the unitarization in the channel c = 1, i.e. due to theaccount for hadronic rescatterings, the singularity of V

(1)confinement(q) may appear in the

ππ-amplitude on the second sheet and one may believe that this singularity is what wecall ”the light σ-meson”.

Therefore, the main question consists in the following: the V(1)confinement(q

2), does

it have the same singular behaviour as V(8)confinement(q

2)? The observed linearity of the(n,M2)-trajectories, up to the large-mass region, M ∼ 2000 − 2500 MeV, see Figs. 1

and 2, favours the idea of the universality in the behaviour of potentials V(1)confinement

and V(8)confinement at large r, or small q. To see that (for example, in the process γ∗ →

qq, Fig. 11a) let us discuss the colour neutralization mechanism of outgoing quarksas a break of the gluonic string by newly born qq-pairs. At large distances, (2.0–2.5 fm), that corresponds to the formation of states with masses 2000–2500 MeV,two–three new qq-pairs should be formed. This follows from the value 〈nch(W )〉 in thee+e−-annihilation: at W ≃ 2 GeV we have 〈nch〉 ≃ 3. It is natural to suggest thata convolution of the quark–gluon combs governs the interaction forces of quarks atlarge distances, see Fig. 11bc. This means that the potential V

(c)confinement working at

such large distances contains two or three t-channel qq-pairs. The mechanism of theformation of new qq-pairs to neutralize colour charges does not have a selected colourcomponent, see e.g. [89]. In this case all colour components 3 ⊗ 3 = 1 + 8 behavesimilarly, that is, at small q2 the singlet and octet components of the potential areuniformly singular, V

(1)confinement(q

2) ∼ V(8)confinement(q

2) ∼ 1/q4. This is seen in Fig. 11a:the quark–gluon ladder ensures the t-channel flow of colour charge C = 3, so quark–antiquark interaction block being the convolution of ladder diagrams 3 ⊗ 3 = 1 + 8contains two equivalent, singlet and octet, components. This points to a similarity ofV

(1)confinement and V

(8)confinement.

So, from the point of view of mechanisms of colour neutralization related to the ideasof the t-channel production of new quark–antiquark pairs, the existence of the lightσ-meson is well-grounded. Therefore, it is of utmost importance to discover whetherthere exist singularities at low energies in the 00++-amplitude, which we call the lightσ-meson, and what is the type of these singularities, if any.

38

4 Conclusion

Recent analysis of meson spectra in the pp-annihilation in flight [6] resulted in theopportunity to systematize qq-mesons, practically leaving too small a room for spec-ulations about the existence of exotic states. First of all, this is related to the 00++

sector, where all comparatively narrow resonances f0(980), f0(1300), f0(1500), f0(1750)lay on linear qq-trajectories on the (n,M2)- and (J,M2) planes, that does not provideus with a ground for hypotheses on their exotic origin.

In quark models of 60–70’s the idea of dominant LS-splitting was pushed forward[90, 91] — within such a structure of forces the P -wave qq-multiplet with J = 0 islighter than multiplets with J = 1, 2. Recently the models became popular, wheref0(980) and a0(980) are considered as exotic states (see [92, 93, 94] and referencestherein): in this case the 13P0qq-nonet should be placed in the region 1300–1700 MeV.It is seen that the K-matrix analysis [8,11,12,13] renders us back to a former picture:the 13P0qq multiplet is the lightest one among the P -wave qq-states.

Beyond the 00++ sector, the candidate for the exotics is the π2(1880) [39] — forsure this state does not belong definitely to the qq-trajectory, and it could be a hybridstate. Of course, the existence of this resonance should be confirmed by its observationin other reactions. Besides, if π2(1880) was the qqg-state, there should exist the othernonet members near 1900 MeV in addition to those that lay on standard (n,M2)trajectories. However, no 2−+ mesons have been observed yet, which could be extraones with respect to the qq classification.

In the literature, there are indications to a plausible existence of the non-quark-antiquark mesons with JPC = 1−+ (see [19] and references therein), though experimen-tal evidence for these states is rather poor.

In the 00++ meson sector, two states are beyond the qq systematics, namely, thelight σ-meson and the broad state f0(1200− 1600).

Concerning the problem of the light σ-meson, though important and having a longdiscussion history, there is no direct evidence of its existence — on the level of fixing apole in the 00++ amplitude on the basis of the experimental data with large statistics.Such a situation existed a few years ago, see the discussion in [76], yet there is nonoticeable experimental progress till now.

Nevertheless there are reasons which make us to concern about the problem ofthe light σ-meson: this is singular behaviour of the quark–antiquark interaction blockat small momentum transfers, 1/q4, in the coordinate space this corresponds to thelinear growth of the potential, ∼ αr, at large r (in quark models such a potential isconventionally called ”confinement potential”). Linear behaviour of meson trajecto-ries in the (n,M2)-plane at large masses (Figs. 1,2) tells us that such a behaviouroccurs actually at rather large r, up to r ∼ 2.0 − 2.5 fm. Assuming that both colourcomponents of the ”confinement potential”, octet and singlet ones, behave at larger similarly, V

(1)octet(r) ∼ V

(8)singlet(r), we gain singular behaviour at small masses in the

39

white (hadronic) channel. Hadron unitarization of the amplitude (the account for theππ-rescatterings), which is necessary in the white channel but is absent in colour one,is capable to ”hide” the white-channel singularity under the ππ branching cut thusre-creating the picture of the light σ-meson. The mechanism of colour neutralization,where the t-channel formation of new qq pairs is of principal importance, favours thehypothesis about similar growth of potentials V

(1)confinement(r) ∼ V

(8)confinement(r).

As concern the broad state f0(1200−1600), we know neither its mass nor the widthreliably. Still, we do know that this broad state is strongly produced in a number ofreactions. Also we know that, in terms of the qq and gluonium components, thisstate is for sure the mixture of the gluonium with the quark-antiquark state whichis close, by its content, to the flavour singlet (qq)glueball = (nn

√2 + ss

√λ)/

√2 + λ

with λ ≃ 0.5− 0.8: this fact is proved rather confidently by the relations between thecouplings f0(1200− 1600) → ππ,KK, ηη, ηη′, which are reliably found from the data.

Another reliably determined feature of the broad state f0(1200 − 1600) is that itis a more loosely-bound system than the surrounding f0-resonances; this fact pointsto a possibility to create it as a result of the accumulation of widths of neighbouringresonances. If so, and the K-matrix analyses of the 00++ waves [8, 9] performed on anextended data basis favours this scenario, then this circumstance explains us where thelightest scalar glueball vanishes, though it promised to be around ∼ 1500MeV but it isdefinitely missing among the observed narrow f0-mesons. In this way, let us emphasizethat the possibility for a new comparatively narrow 00++ resonance near 1000-1800MeV is strictly excluded by the available experimental data.

A scenario for f0(1200−1600) being the glueball descendant, which lost a part of itsgluonium component due to the mixing with neighbouring resonances, was suggestedin [68, 69], on the basis of data which were much poorer than the present ones. Thenowaday data which are richer not only in statistics but also in a number of investigatedreactions, fit to this picture rather well, without noticeable contradictions to it. Onemore check of the picture with f0(1200 − 1600) as a glueball descendant is expectedin the nearest future for the high-statistics radiative decays J/Ψ → γ + hadrons: inhadronic spectra, the production of comparatively narrow resonances f0(1300) andf0(1500) in the 00++ wave should be accompanied by a strong ”background” due tothe production of the broad state f0(1200− 1600).

At the time being the mixing of f0-mesons at 1300–1700 MeV is intensively dis-cussed, see recent papers [95, 96] as an example. However, it should be speciallystressed that the mixing considered in the K-matrix technique is in principle differentfrom mixing discussed in [95, 96]. In the K-matrix approach, the mixing occurs dueto transitions of bare states to real mesons — in other words, due to imaginary partsof hadron loop diagrams. As a result of such a mixing the amplitude poles ”move” inthe complex mass plane, a loss or accumulation of widths take place, that is, mixedstates ”push” each other in the complex plane by sinking deeply into complex planeor springing out to real axis; as a result, real mesons-resonances are created. In thisway the shift of real part of resonance mass may be considerable — of the order of

40

resonance width. The mixing discussed in the papers [95, 96] happens without decayprocesses, that is, without imaginary parts of hadronic loop diagrams, so correspond-ing amplitude poles are on real axis: by mixing the levels repulse each other, with theincrease or decrease of the mass but not width; after such a mixing the bare statesshould be created but not real resonances, as was suggested in papers [95, 96]. Onceagain: in the K-matrix technique the formation of the observed resonances takes placeby the onset of the decay processes only.

In accordance with gluodynamical lattice calculations, the second scalar glueballshould exist at 2100–2200 MeV [97]. In [98], it was suggested that f0(2105) is eitherthis second glueball or strongly mixed with it. However there is not enough data tojudge about the glueball being in this mass region: the discovered resonances lay on theqq trajectories quite comfortably. It is rather probable that the second scalar glueballturned into the broad state too by mixing with the neighbouring qq mesons — in thiscase to fix it experimentally the measurement of a large variety of spectra is neededas well as the evaluation of ratios for the ππ, KK, ηη, ηη′, η′η′ yields in the region2100–2200 MeV.

Acknowledgement

I am indebted to A.V. Anisovich, Ya.I. Azimov, D.V. Bugg, L.G. Dakhno, Yu. S.Kalashnikova, D.I. Melikhov, V.A. Nikonov and A.V. Sarantsev for helpful discussionsof problems involved. This work is supported by the RFBR Grant N 0102-17861.

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Figure 1: Trajectories on the (n,M2) planes for the states with (C = −). Open circles stand for thepredicted states.

47

Figure 2: Trajectories on the (n,M2) planes for the states with (C = +).

48

Figure 3: (J,M2) planes for leading and daughter trajectories: a) π-trajectories, b) a1-trajectories,c) ρ-trajectories, d) a2-trajectories, e) η-trajectories, f) P

′-trajectories.

49

Figure 4: ComplexM plane in the (IJPC = 00++) sector. Dashed line encircle the partof the plane where the K-matrix analysis [8] reconstructs the analytic K-matrix am-plitude: in this area the poles corresponding to resonances f0(980), f0(1300), f0(1500),f0(1750) and the broad state f0(1200 − 1600) are located. On the border of this areathe light σ-meson denoted as f0(450) is shown (the position of pole corresponds tothat found in the N/D method [31]). Beyond the K-matrix analysis area, there areresonances f0(2030), f0(2100), f0(2340) [6].

Figure 5: Complex M plane: trajectories of poles corresponding to the states f0(980), f0(1300),f0(1500), f0(1750), f0(1200− 1600) within a uniform onset of the decay channels.

50

Figure 6: The evolution of normalized coupling constants γa = ga/√∑

b g2b at the onset

of the decay channels for f0(980), f0(1300), f0(1500), f0(1750). Curves demonstratethe description of constants by formula (38).

Figure 7: (a,b) Examples of planar diagrams responsible for the decay of the qq-state and gluoniuminto two qq-mesons (leading terms in the 1/N expansion).

51

Figure 8: Correlation curves on the (ϕ,G/g) and (ϕ, g/G) plots for the description ofthe decay couplings of resonances (Table 4) in terms of quark-combinatorics relations(38). a,c) Correlation curves for the qq-originated resonances: the curves with appro-priate λ’s cover strips on the (ϕ,G/g) plane. b,d) Correlation curves for the glueballdescendant: the curves at appropriate λ’s form a cross on the (ϕ, g/G) plane with thecenter near ϕ ∼ 30◦, g/G ∼ 0.

52

Figure 9: Linear trajectorieson the (n,M2) plane for scalar resonances (a) and bare scalar states(b). Open circles correspond to the predicted states.

53

Figure 10: The (JP ,M2) planes for kaonic sector (open circles stand for the predicted states).

54

a b

Figure 11: a) Quark-gluonic comb produced by breaking a string by quarks flowingout in the process e+e− → γ∗ → qq → mesons. b) Convolution of the quark-gluoniccombs — an example of diagrams describing interaction forces in the qq systems atr ∼ 2.0 fm.

55